Technology Toolkit (TI-89)

Technology Toolkit (TI-89)

D A V I D T Y N A N

•

J O H N D O W S E Y

•

L Y N D A B A L L

MathsWorld

Contents

1

Home screen calculations

1.1 How to perform simple arithmetic calculations

1.2 How to store and use numerical values

1.3 How to store and use lists

1.4 How to perform simple function calculations

1.5 How to work with angles

1.6 How to perform simple trigonometric calculations

1.7 How to perform simple statistical calculations

1.8 How to perform simple probability calculations

1.9 How to perform simple symbolic calculations

2

Representing functions

2.1 How to enter and plot functions

2.2 How to create a table of values

2.3 How to ‘jump to’ significant points on a graph

2.4 How to find the intersection point of two graphs

3

Moving around the viewing window

3.1 How to change the viewing window

3.2 How to zoom options

4

Advanced function graphing features

4.1 How to enter and plot a function using parameters

4.2 How to shade above/below a function graph

4.3 How to graph functions defined in terms of other functions

4.4 How to draw the inverse of a function

4.5 How to restrict the domain of a function

4.6 How to work with hybrid functions

4.7 How to work with reciprocal functions

4.8 How to work with rational functions

4.9 How to work with sum and difference functions

4.10How to work with products of functions

4.11How to work with composite functions

4.12How to work with functional equations

5

Calculus

5.1 How to calculate average and instantaneous rates of change

5.2 How to calculate the numeric derivative

5.3 How to calculate and plot derivative functions

5.4 How to draw tangent lines

5.5 How to calculate the definite integral

5.6 How to find the areas between two curves

5.7 How to find the average value of a function

5.8 How to calculate and use the second derivative

6

Working with data

6.1 How to store and summarise ungrouped univariate data

6.2 How to store and summarise grouped univariate data

6.3 How to construct cumulative frequency curves

6.4 How to construct a histogram

6.5 How to construct a box plot

6.6 How to store and summarise bivariate data

6.7 How to construct a scatter plot

6.8 How to calculate correlation coefficients

6.9 How to apply a regression model

7

Numeric solvers

7.1 How to use the numeric solver APP

7.2 How to use the financial solver APP

8

Working with matrices

8.1 How to store and use matrices

8.2 How to solve equations with matrices

8.3 How to transform points and equations with matrices

8.4 How to work with transition matrices

9

Working with sequences

9.1 How to define, plot and tabulate a sequence rule

9.2 How to sum a sequence

9.3 How to work with difference equations

10

Symbolic commands

10.1How to define and use functions

10.2How to use the symbolic solve command

10.3How to work with general solutions

10.4How to rearrange equations and expressions

10.5How to work with limits

10.6How to find the symbolic derivative

10.7How to calculate indefinite integrals

11

Probability distributions

11.1How to analyse a discrete probability distribution

11.2How to analyse a continuous probability distribution

11.3How to analyse a binomial probability distribution

1.1

How to perform simple

arithmetic calculations

The following assumes that the calculator is at the home screen (press HOME if it is not) and using the default settings, which includes the use of RADIAN mode and the AUTO calculations mode. The display digit is set to FLOAT. Note that it is assumed that

¸

is pressed to complete each command.

Task Keystrokes Result

Evaluate 3.52

3.5^2

12.25 (stays as a decimal)

Evaluate

7 2

2

(7/2)^2

49/4 (stays as a fraction)

Evaluate

1 7

0

2

[

] (10÷7)

Evaluate

489

89^(1÷4)

Evaluate 687,000

6

87000

522000. (stays as a decimal)

Evaluate 6(8.7104)

6

8.7

^

4

522000

Evaluate (3.1106)(2.9104)

3.1

^

6÷2.9

^

4

106.8966

Is 7557 ?

7^5

2

[<] 5^7

True

Note: use

¥

instead of

2

to enter ≤symbol

Find the prime factors of 108 Press

„

(Algebra)

2233

Select 2:factor(

Complete as follows factor(108)

Find log10137 Press

k

2.1367

Scroll to log( and press

¸

Note: the natural log key is

Complete as follows:

2 p

log(137)

¥ ¸

Express 1.6 as a fraction Press

2

m

_

8/5

Select 1:Numberand 1:exact(

89

1 4

70

7

1

Home screen

1.2

How to store and use numerical values

Numerical calculations are made which give results that may be stored and used in later calculations. As the following examples show, these can be stored in memory as the alphabetic letters. Remember to use

j

when using letters. Note that uppercase letters are displayed on the keys, but lowercase letters are displayed on the screen.

1.3

How to store and use lists

Task Keystrokes Result

Calculate

23

and store it as d

2

(

)(23)

§

d

23

Calculate the value of 3(13 – d)2

3(13–d)^2

§

q

6(13

23

96) and store it as q

Evaluate V r2h when r= 3

3

§

r

and h= 2 and store the

2

§

h

answer as V

2

π

r^2

h

§

v

(or all in one line separated by colons – see screen)

Evaluate V r2h

2

π

r^2

h

Í

r=3 and h=2

when r= 3 and h= 2 (NB ‘and’ accessed via

k

and the

Í

key is below the

Á

key)

Task Keystrokes Result

Store the numbers 25, 72 and 56

{25,72,56}

§

z

{25 72 56}

as a list called z

Find the difference between Press

k

and locate

Δ

list

{47 –16} successive values in list z Complete as follows:

Δ

list (z)

Find the prime factors of the Press

„

(Algebra)

{52 2332 237} numbers of the list z Select 2:factor(

Complete as follows: factor(z) or

factor({25,72, 56})

Generate the first 4 terms of the

seq(1000

(0.8)^(n–1),n,1,4)

{1000. 800. 640. 512.}

sequence with rule

§

T

t_n1000(0.8)n1and store this in t

Graph y , y and y Press

¥

[Y=]

on the same axes Type y1=x÷{2,3,4}

Press

„

and select 4:ZoomDec

x

4

x ₃ x

1.4

How to perform simple function

calculations

1.5

How to work with angles

Unless otherwise stated, it is assumed that the TI-89 is in RADIAN mode.

Task Keystrokes Result

Evaluate x22x1 when x = 5

x^2+2x–|x=5

34

Evaluate yx22x1 when Type x^2+2x–1

§

y(x)

34

x = 5 Type y(5)

Define a function A(r) r2 Press

†

(Other), select 1:Define

Type Define a(r)=

π

r^2

Done

Calculate the value of A(r)

a(17)

289

for r17cm

Calculate the value of A(r)

a({1,2,3})

{ 4 9 }

for r1,2,3 cm

Calculate the value of A(r) if r

a({2r,3r,4r})

{4 r2 9 r2 16 r2} is doubled, tripled, or quadrupled

Task Keystrokes Result

Express 56.35° in degrees,

56.35

2

[°]

minutes, seconds Press

2

[MATH]

Select 2: Angleand 8:>DMS 56°21

Express 56°21in decimal degrees

56

2

[°] 21

2

[‘]

56.35° Press

2

[MATH]

Select 2: Angleand 9:>DD

and

¥ ¸

Express 56.35° in radians

56.35

2

[°]

0.983493

Express 56°21in radians

56

2

[°] 21

2

[‘]

Express in degrees and Press

(3

π

÷8)

2

[MATH]

67°30

minutes Select 2:Angleand 8:>DMS

Express 1.26cto degrees, minutes

1.26

2

[MATH]

72° 1133.6559

& seconds Select 2:Angleand 8:>DMS

3

₈

1127

1.6

How to perform simple

trigonometric calculations

1.7

How to perform simple

statistical calculations

Task Keystrokes Result

Evaluate sin(36°24) Press

2

[SIN] (36

2

[°] 24

0.5934 (to 4 d.p.)

2

[‘] )

then

¥ ¸

Evaluate cos

Press

2

[COS] (2

π

/5)

0.3090 (to 4 d.p.)

and

¥ ¸

Evaluate tan(27.34°) Press

2

[TAN] (27.34

2

[°])

0.5170 (to 4 d.p.)

and

¥ ¸

Find θin radians if cos θ= 0.3 Press

¥

[COS

–1

] (0.3)

1.2661 (to 4 d.p.)

Find θin degrees, minutes and Press

¥

[TAN

–1

] (1.7)

59 324

seconds if tan θ= 1.7 Press

2

[MATH]

Select 2:Angle and 8:>DMS 2

5

Task Keystrokes Result

Find the mean of Press

2

[MATH]

27.3

{26.3, 24.1, 22.4, 36.4} Select 6:Statisticsand 4:mean( Complete as follows:

mean({26.3,24.1,22.4,36.4})

Find the median of Press

2

[MATH]

25.2

{26.3, 24.1, 22.4, 36.4} Select 6:Statisticsand 8:median( Complete as follows:

median({26.3,24.1,22.4,36.4})

Find the standard deviation of Press

2

[MATH]

6.27 (to 2 d.p.) {26.3, 24.1, 22.4, 36.4} Select 6:Statisticsand 6:stdDev(

Complete as follows:

1.8

How to perform simple

probability calculations

Task Keystrokes Result

Evaluate 10!

10

2

[MATH] and select

3628800

7:Probability

and 1:!

8 students are in a 100m race. Press

2

[MATH]

336

How many ways can they be Select 7:Probabilityand 2:nPr placed 1st, 2nd and 3rd? Complete as follows: nPr(8,3)

How many groups of 3 can be Press

2

[MATH]

286

made from 13 people? Select 7:Probabilityand 3:nCr Complete as follows: nCr(13,3)

Seed the random number Press

2

[MATH]

Note: All students should choose

function Select 7:Probabilityand a different seeding number

6:RandSeed

Complete as follows:

RandSeed 89267

Generate a random number Press

2

[MATH]

Answers will vary!

between 0 and 1 Select 7:Probability and 4:rand(

Complete as follows: rand()

Generate a random integer Press

2

[MATH]

Answers will vary!

between 1 and 20 Select 7:Probability and 4:rand(

Complete as follows: rand(20)

Generate a random integer Press

k

and int( Answers will vary!

between 20 and 36 then

2

[MATH]

Select 7:Probability and 4:rand( Complete as follows:

int(rand()

17)+20

Graphics calculators generate ‘random numbers’ based on a sequence of numbers defined by a rule. Because of this, they are not truly random, and are sometimes called ‘pseudo-random’ numbers. When a calculator is reset, it will start at the same point in the sequence as another reset calculator; each

calculator will begin with the same ‘random’ number. To counter this, the random number generator can be seeded as shown above, so that each calculator produces a different sequence of random numbers.

J

1.9

How to perform simple symbolic

calculations

The follow section illustrates the syntax for some simple symbolic computations with the TI-89. It assumes that the calculator is using the default settings.

Task Keystrokes Result

x+x

2 .x

x

x

x2

xx+x

x

x2xx(xxtreated as a

single variable)

a

x + b

x

(x^2–y^2)÷(x–y)

2(y–x)

(x–y)

(x–y)

(x–y)

(x

y)

(x^2

y^3)^–2

Divide by Press

„

(Algebra)

Select 7:propFrac( Complete as follows:

propFrac((3x–2)÷(x+1))

if r= 2 cm (exactly)

2

[

π

]

r^2

Í

r=2

if r= 2 cm (2 d.p.)

2

[

π

]

r^2

Í

r=2

¥ ¸

12.5664

if r= h + 3

2

[

π

]

r^2

Í

r=h+3

Expand Press

„

(Algebra)

Select 3:expand( Complete as follows:

expand((x+y)^3)

Factorisex29 Press

„

(Algebra)

Select 2:factor( Complete as follows:

factor(x^2–9)

Is x22xx(x2)

x^2 + 2x = x

(x+2)

true

(x −3) (⋅ x+3)

x3 +3⋅x2 ⋅y +3⋅ ⋅x y2 + y3

(x+ y)3

(h+3)2 ⋅π

πr2

πr2

4⋅π

πr2

3 5 1

−

+

x x +1

3x −2

1

4 6

x ⋅y x y2 3 2

( )

−

(x+ y) (⋅ x −y) (x −y x)( + y)

(x− y)2 (x −y x)( −y)

− ⋅2 (x −y) 2(y −x)

x + y

( ) ( )

x y

x y

2 ₋ 2 −

(a+b)⋅x a× + ×x b x

Task Keystrokes Result

Factorise Press

„

(Algebra)

Select 2:factor( (factors over rational numbers)

Complete as follows:

factor(x^2–3)

Factorise over R Press

„

(Algebra)

Select 2:factor(

Complete as follows: (factors over real numbers)

factor(x^2–3,x)

Express 2

x

3

yas a single Press

„

(Algebra)

fraction Select 6:comDenom(

Complete as follows:

comDenom(2÷x+3÷y)

Solve for x Press

„

(Algebra)

Select 1:solve( Complete as follows:

solve(x+y–5=2x–5y,x)

Solve for x Press

„

(Algebra)

x3 or x 3

Select 1:solve( Complete as follows:

solve(x^2–9=0,x)

Solve for xif x > 0 Press

„

(Algebra)

Select 1:solve( Complete as follows:

solve(x^2–9=0,x)

Í

x>0

Solve for h Press

„

(Algebra)

Select 1:solve( Complete as follows:

solve(v=

2

[

π

]

r^2

h,h)

Solve 2x6 for x

solve(2x<6,x)

x3

Solve x230 for x

solve(sign(factor(x^2-3,x))

= -1,x)

x

3

and x

3

Note: The ‘sign’ function tests whether an equation is positive or negative, and returns +1 if positive and –1 if negative. This function can be accessed via the

k

key. h v r = ⋅ π 2

V = πr h2

x = 3

x2 −9 = 0

x2 −9 = 0

x = 6⋅ −y 5

x + y −5 = 2x−5y

3⋅ + 2⋅

⋅

x y

x y

x + x

(

3

)

⋅

(

− 3

)

x2 −3

x2 −3

2.1

How to enter and plot functions

The usual way to graph a function is to use the Y=editor. Functions can be entered directly (for example, y1=x^2) or indirectly (for example,

y2=1+y1(x)

where y1 is already defined, or y3=f(x)where f(x) has been previously defined at the home screen). The independent variable for graphing must always be xand the graph mode must be set to FUNCTION. (Press MODE to check)

Alternatively, it is possible to graph functions directly from the home screen using the Graphcommand in the

†

(Other)

menu. This method allows you to specify an expression in terms of any independent variable. However, it is generally preferable to use the Y=editor.

E

E x a m p l e 1

Graph the following in an appropriate viewing window:

a y= x2

b y= 2x2

c y= x2– 2, y = x2– 1, y = x2+ 1, y = x2+ 2

Solution

a Press

¥

[Y=]

Type y1=x^2

¸

Press

¥

[GRAPH]

to view the graph.

The view is dependent on the window dimensions used previously. To use the standard viewing window (–10 ≤x≤10 and –10 ≤y≤10):

Press

„

(Zoom), select 6:ZoomStd

Press

…

(Trace)

and

A

or

B

to trace points.

(The 1 in the top right corner indicates tracing the graph of y1.)

2

Representing

b When one function is related to another, it can often be useful to express that relationship in the rule.

Press

¥

[Y=]

Type y2=2

y1(x)

¸

Press

¥

[GRAPH]

to view the graph.

To trace the graphs, press

…

(Trace)

and

C

or

D

to move between them.

c The four rules can be entered together to show their relationship to the graph of y= x2.

First turn off the plot of y2 in the following manner: Press

¥

[Y=]

and place the cursor at y2.

Press

†

to deselect the graph of y2. (Note that the rule is not cleared, but its graph will not be plotted.)

Type y3=y1(x)+{–2,–1,1,2}

Press

¥

[GRAPH]

to view the graph and trace as before.

E

E x a m p l e 2

Graph the function fwith rule f(u) = 3u2– 2u– 1.

Solution

To graph a function rule with an independent variable other than x: Press

t

to display the home screen.

Press

†

(Other), select 2:Graph

Type Graph 3

u^2–2

u–1,u

2.2

How to create a table of values

The following procedure will produce a table for a specified function rule. It assumes that the calculator is using the default settings.

E

E x a m p l e

For the function fwith rule f(x)x(x3)(x2), tabulate f(x) for

Solution

Press

¥

[Y=]

Type y1=x

(x+3)

(x–2)

Press

„

(Zoom), select 6:ZoomStd

Press

¥

[TblSet]

For the ‘tblStart’ value, type 2 For the ‘Δtbl’ value, type 2 Press

¸

to save the settings.

Press

¥

[TABLE]

The values of are displayed. Pressing

C

and

D

displays table values outside the specified set of xvalues.

2.3

How to ‘jump to’ significant

points on a graph

Important points on a graph include the intercepts and turning points. It is useful to be able to jump to a point on a graph for a given x-value and to find points on the graph corresponding to a given y-value. This can be achieved interactively on the graph screen if there is a good view of the graph.

E

E x a m p l e

For the graph of y= 0.1x3– 0.2x, find:

a the y-coordinate at x= 1.2

b any intercepts

c the coordinates of any turning points.

f x( )

Solution

a Press

¥

[Y=]

Type y1=0.1

x^3–0.2

x

Press

¥

[WINDOW]

Change the window dimensions to [–3, 3] by [–0.5, 0.5]. Press

¥

[GRAPH]

to view the graph in this window.

To jump to the graph at x= 1.2: Press

…

(Trace)

Type 1.2

The corresponding y-value is –0.0672. The cursor jumps to the pixel on the screen closest to that point.

b There are three intercepts, one of which appears to be at x = 0. Verify this by jumping to x= 0 as explained above. This also shows that the y-intercept is 0.

To find the leftmost x-intercept: Press

‡

(Math), select 2:Zero

There will be a prompt for the estimated x-coordinates between which a ‘zero’ for y1will occur:

For ‘Lower Bound’, type –2 For ‘Upper Bound’, type –1 (Alternatively, press

A

and

B

.)

The x-intercept is at x= –1.414 (correct to 3 d.p.). Repeat the procedure to find the rightmost x-intercept (which is at x= 1.414).

c There are two turning points. To find the leftmost: Press

‡

(Math), select 4:Maximum

There will be a prompt for the estimated x-coordinates between which a local maximum for y1will occur:

For ‘Lower Bound’, type –2 For ‘Upper Bound’, type 0 (Alternatively, press

A

and

B

.)

The maximum is located at (–0.816, 0.109).

A similar procedure is used to find the local minimum: Press

‡

(Math), select 3:Minimum

As well as the graphical methods shown here, the TI-89 has symbolic commands such as zeros, fMinand

fMax wich may help locate the coordinates at significant points.

J

2.4

How to find the intersection

point of two graphs

It is often necessary to find the intersection points of two or more graphs. While features such as ZoomBoxand

ZoomIn

enable close inspection of such points, the intersection feature can quickly find the coordinates of any intersection points.

E

E x a m p l e

For the cubic y= 0.5x3– 2x+ 3 and the parabola y= 7 + 2x– x2:

a plot the graphs of both functions in a suitable window

b find the coordinates of any intersection points

c find the x-values on the cubic graph corresponding to y= 2.

Solution

a Press

¥

[Y=]

Type y1=0.5

x^3–2

x+3

Type y2=7+2

x–x^2

Press

„

(Zoom), select 6:ZoomStd

In the standard window, two intersection points can be seen and it looks like there is a third outside the viewing window.

Press

¥

[WINDOW]

and adjust the window settings as follows:

xmin = –5

xmax = 5

ymin = –20

Leave the other settings unchanged and press

¥

[GRAPH]

Now all three intersection points can be seen.

b To find the leftmost intersection point: Press

‡

(Math), select 5:Intersection

There will be a prompt for the curves whose intersection point you are interested in:

For ‘1st curve’, place the cursor on y1and press

¸

For ‘2nd curve’, place the cursor on y2and press

¸

There will be a prompt for the estimated x-coordinates between which the desired intersection occurs:

One of the intersection points is located at (–3.604, –13.20).

Repeat the method to find the other two intersection points: (−0.890, 4.428) and (2.494, 5.768).

c Press

¥

[Y=]

Type y3=2

Press

„

(Zoom), select 4:ZoomDec

The horizontal line y= 2 intersects the cubic three times (and the parabola twice). Use the method described in part bto find the coordinates of each intersection with the cubic.

3.1

How to change the viewing

window

Many graphs will not show all key features or may not even appear in the standard viewing window. You can adjust the window settings or use the Zoom features as needed.

The following examples illustrate some of these features.

E

E x a m p l e

Graph the functions with rules:

a y= x2+ 10x– 6

b y= 0.1x3– 0.2x

Solution

a Press

¥

[Y=]

Type y1=x^2+10x

−

6

Press

„

(Zoom), select 6:ZoomStd

to view the graph in the standard [−10, 10] by [−10, 10] window.

Observe that such features as intercepts, the local minimum and graph shape are not all shown in this window.

To get a better view of such features: Press

„

(Zoom), select 3:ZoomOut

Move the cursor to a new centre near (–5, 0) and press

¸

The parabola now appears with all its key features.

3

Moving around the

To specify a particular viewing window, use the Window Editor: Press

¥

[WINDOW]

and type the required values, for example:

xmin = –15

xmax = 5

xscl = 1

ymin = –35

ymax = 50

yscl =1

Press

¥

[GRAPH]

to view the graph in this window.

b Press

¥

[Y=]

Type y1=0.1

x^3–0.2

x

Press

„

(Zoom), select 8:ZoomInt

to enable integer tracing. (Note: If the tracing is in odd amounts, press

¥

[WINDOW]

and type

xres=1. Press

¥

[GRAPH]

and

…

(Trace)

ZoomBox

gives a better view of the graph near the origin. The box area you specify becomes the new viewing window.

Press

„

(Zoom), select 1:ZoomBox

Use the arrow keys to move the cursor to (–3, 1) and press

¸

Use the arrow keys to move the cursor to (3, –1) and press

¸

3.2

How to use zoom options

Zoom settings are either special windows, or ways of achieving useful windows in particular contexts. A brief explanation of each is given below.

Zoom setting What it does

ZoomBox

Lets you draw a box and zoom in on that box. Move the cursor to locate one box corner and press

¸

, then locate the opposite box corner and press

¸

. This box becomes the new viewing window.

ZoomIn &

Lets you select a point and zoom in or out by an amount defined by SetFactors (default

ZoomOut

factor = 4). It first prompts the user to relocate the centre point (if desired). When

¸

is pressed, the zoom step is performed.

ZoomDec

Sets the xand yincrements (Δxand Δy) to 0.1 which creates the same scale on each axis, and centres the origin. Always creates the window [–7.9, 7.9] by [–3.8, 3.8].

ZoomSqr

Adjusts window variables so that a square or circle is shown in correct proportion (instead of a rectangle or ellipse, respectively).

ZoomStd

Sets window variables to their default values, which are: xmin = –10

xmin = –10 and

xres = 2 (the number of pixel columns between calculations) xmax = 10

ymax = 10 xscl = 1

yscl = 1 (the space between axis ticks)

ZoomTrig

Graphing trigonometric functions is made easier by the careful use of the window dimensions, including consideration of the xscl setting, which will set the space between ticks on the x-axis. The window setting ZoomTrig may also be helpful, which creates a

2 7 4 9

, 7 2 9 4

by [4, 4] viewing window of the function with a helpful trace increment of

24.

ZoomInt

Lets you select a new centre point, and then sets Δxand Δyto 1 and sets xscl and yscl to 10.

ZoomData

Adjusts window variables so that all selected stat plots are in view.

ZoomFit

Adjusts the viewing window to display the full range of dependent variable values for the selected functions. In function graphing, this maintains the current xmin and xmax and adjusts ymin and ymax.

Memory

Lets you store and recall window variable settings so that you can recreate a custom viewing window. ZoomPrevious is like an UNDO – it restores the previous Zoom setting.

4.1

How to enter and plot a

function using parameters

For a function defined in terms of a parameter, for example, f(x) = x+ cwhere

cis a real number, we may want to know how the graph of y= f(x) changes as the parameter cchanges. A useful technique is to store a set of numerical values to the parameter in the form of a parameter list. They can be entered directly (i.e. (y1x{–2,–1,0,1,2}) or by storing the values as c

(i.e. {–2,–1,0,1,2}

§

c) at the home screen and entering the rule y1=x+c.

The following illustrates the latter approach

E

E x a m p l e

Let y= mx+ cwhere mand care integers. How do the straight lines change in the following cases?

a m= 1 and cvaries

b c= 0 and mvaries

Solution

a Press

t

Type 1

§

m

¸

Type {–2,–1,0,1,2}

§

c

¸

Press

¥

[Y=]

Type y1=m

x + c

Press

„

(Zoom)

and select 4:ZoomDec

As expected, the graphs are straight lines with gradient 1 and changing

y-intercepts. Trace the graphs in the usual way: the trace starts on the first graph in the list (yx2).

4

Advanced function

b Press

t

Type 0

§

c

¸

Type {–2,–1,0,1,2}

§

m

¸

Press

¥

[GRAPH]

As expected, the graphs are straight lines through the origin with changing gradients. Trace the graphs in the usual way: the trace starts on the first graph in the list (y= –2x).

4.2

How to shade above/below a

function graph

The TI-89 has four shading patterns, used on a rotating basis. If you set one function as shaded, it uses the first pattern. The next shaded function uses the second pattern, etc. The fifth shaded function reuses the first pattern. When shaded areas intersect, their patterns overlap.

E

E x a m p l e

Shade the region given by the following inequalities:

a 2x+ y4

b x– y 1

c represent the region given by the two above inequalities and x 0 and

y0.25 (this time, for clarity’s sake, leave the feasible region unshaded).

Solution

a The inequality 2x+ y4 can be written as y –2x+ 4. Press

¥

[Y=]

Type y1=–2x+4

¸

Press

ˆ

(Style), select 8:Below

b The inequality x– y 1 can be written as yx – 1. Press

¥

[Y=]

Type y1=x – 1

Press

ˆ

(Style)

and select 7:Above

Press

„

(Zoom)

and select 4:ZoomDecto view the inequality graph. Note that the region given by the inequality is shaded (above the line).

c Enter the following in y1, y2 and y3 as shown. Note that x= 0 is not possible to graph here.

Use the following styles (remember now we are going to shade the region that does not satisfythe inequality):

y1= –2x+4

(shade above)

y2=x

– 1

(shade below)

y2= 0.25

(shade below)

To represent the feasible region, Press

¥ $

. Change the viewing window to [0,2] by [0,3.8].

Note that setting xmin = 0 allows us to ‘represent’ the region x≥0. In this case, the feasible region has been represented as the ‘unshaded’ region.

4.3

How to graph functions defined

in terms of other functions

For any given function and its graph, there may be one or more

transformations, such as:

p reflection in a coordinate axis

p translation horizontally or vertically, or

p dilation (that is, stretching) parallel to the coordinate axes.

E

E x a m p l e

For the function with rule f(x) = 1 + 2x– x2, plot the graph of y= f(x) and explain graphically the effect of the following transformations:

a –f(x) b f(–x) c f(x– 3)

d f(x) – 3 e f(2x) f 2f(x)

Solution

To enter a rule as f(x):

Press

t

to display the home screen. Press

†

(Other), select 1:Define

Type Define f(x)=1+2

x–x^2

¸

To plot the graph of y= f(x):

Press

¥

[Y=]

Type y1=f(x)

Press

2

[F6], select 4:Thick

Press

„

, select 4:ZoomDec

a Press

¥

[Y=]

Type y2=–f(x) Press

¥

[GRAPH]

The effect is a reflectionof the first graph about the x-axis.

b Press

¥

[Y=]

Type y2=f(

−

x)

Press

¥

[GRAPH]

The effect is a reflectionof the first graph about the y-axis.

c Press

¥

[Y=]

Type y2=f(x

−

3)

Press

¥

[GRAPH]

d Press

¥

[Y=]

Type y2=f(x)

−

3

Press

¥

[GRAPH]

The effect is to produce a translationof the first graph by 3 units in the negative ydirection, that is, to move the graph 3 units down.

e Press

¥

[Y=]

Type y2=f(2x) Press

¥

[GRAPH]

The effect is to produce a dilationof the first graph by a factor of 1/2 parallel to the x-axis, that is, the graph is squashed in the horizontal direction to half its width.

f Press

¥

[Y=]

Type y2=2f(x) Press

¥

[GRAPH]

The effect is to produce a dilationof the first graph by a factor of 2 parallel to the y-axis, that is, the graph is expanded in the vertical direction to twice its height.

4.4

How to draw the inverse of a

function

Obtaining the inverse function f–1of a given function on a given domain involves finding the range of the given function and deriving the rule for the inverse. For an inverse function to exist, the given function must be one-one (so that for each xvalue, there is exactly one yvalue and vice versa). Graphing the given function and its inverse on the same axes can be quite revealing.

E

E x a m p l e 1

The function f is defined by f(x) = e0.5x– 1for all real values of x.

a Plot the graph of f.

b Find the inverse function f –1.

c Plot the graph of the function and its inverse on the same set of axes in a suitable window.

Solution

a Press

¥

[Y=]

Type y1=e^(0.5x–1)

¸

xmax = 4

ymin = –2

ymax = 2

Press

¥

[GRAPH]

The function is one-one with range R+. So, an inverse function exists with domain R+.

b To find the rule for the inverse, swap xand yin the rule for the given function and solve for y, that is, let x= e0.5y– 1and find y.

Press

t

Press

„

, select 1:solve(

Type solve(x=e^(0.5y–1),y)

¸

The inverse has rule y= 2(loge(x) + 1) and the calculator is able to give the correct domain in this case. So, the inverse function is f –1where

f –1(x) = 2(loge(x) + 1) for x>0.

c Press

¥

[Y=]

Type y2=2(

2

[ln] (x)+1)

Type y3=x

Place the cursor at y3.

Press

ˆ

(Style), select 2:Dot

Press

„

(Zoom), select 5:ZoomSqr

ZoomSqr

gives a true scale: the xand yincrements are the same actual size. Graphing y= xon the same axes as the function and its inverse shows that they are reflections of each other in the line y= x. To get a better window:

Press

¥

[WINDOW]

and change ymaxto 6. Press

„

(Zoom), select 5:ZoomSqr

The graphs intersect each other twice on the line y= x.

E

E x a m p l e 2

The function f is defined by f(x) = for x≥–2.

a Graph the function and then draw its inverse on the same set of axes in a suitable window.

b Find the inverse function f–1.

c Graph the function and its inverse on the same set of axes in a suitable window.

Solution

a Press

¥

[Y=]

Type y1=–1+

2

[

√

] (2x+4)

Press

„

(Zoom),

select 4:ZoomDec

The graph appears to be half a parabola on its side, and we can see that the function is one-one. As f(–2) = –1, the function has range [–1, ∞). So,

its inverse exists for [–1, ∞).

We can draw the inverse as follows: Press

ˆ

(Draw), select 3:DrawInv

Type DrawInv y1(x)

¸

This shows the shape of the inverse, but it is just a drawingwhich cannot be traced or used. Nor do we get the rule for the inverse.

b To find the rule for the inverse, swap xand yin the rule for the given function and solve for y, that is, and find y.

Press

t

Press

„

, select 1:solve(

Type solve(x=–1+

2

[

√

] (2y+4),y)

¸

The inverse has rule y= x 2

2

+ x– 3

2 and the TI-89 is able to give the correct domain in this case. So, the inverse function is f –1where

f–1(x) = x 2

2

+ x– 3

2for x≥–1.

c Press

¥

[Y=]

Type y2=0.5

x^2+x–1.5

Í

x

≥

–1

¸

Type y3=x

¸

Move the cursor to y3, press

ˆ

(Style)

and select 2:Dot Press

¥

[GRAPH]

Note the importance of including the domain restriction, otherwise a complete parabola and not the inverse will be displayed.

4.5

How to restrict the domain of a

function

Sometimes we need to restrict the domain of a function and consider its behaviour in a restricted domain. This can be done using the Define command together with

Í

.

E

E x a m p l e 1

Define the function f(x) = x2– 2xfor:

a x>0

b –1 ≤x≤5.

In each case, evaluate the function for x= –5, 0, 5 and 10, and plot the graph of the function.

Solution

a Press

t

Press

†

(Other), select 1:Define

Type Define f(x)=x^2–2

x

Í

x>0

¸

To evaluate f(–5), f(0), f(5) and f(10): Type f({–5,0,5,10})

¸

(With functions defined this way, the {} method of evaluation fails. Each point must be evaluated separately.)

To evaluate each point separately:

f(–5)

f(0)

f(5)

f(10)

The results confirm that the domain restriction is working. To plot the graph, press

¥

[Y=]

Type y1=f(x)

b Press

t

Press

†

(Other), select 1:Define

Type Define f(x)=x^2–2

x

Í

x

≥

0 and x

≤

5

¸

To evaluate f(–5), f(0), f(5), f(10):

f(–5)

f(0)

f(5)

f(10)

To plot the graph in a suitable window: Press

¥

[WINDOW]

Type ymax=20

¸

Press

¥

[GRAPH]

E

E x a m p l e 2

Some functions have different rules for different subsets of their domain. They are sometimes called hybrid functions or piecewise defined functions. The

when

function can be used to define and plot the graphs of hybrid functions. The function f is defined by

a Enter the hybrid rule and evaluate the function for x= –5, 0, 5, 10.

b Plot the graph of the function.

Solution

a Press

¥

[Y=]

Type y1=when(x<0,–x,x^2+1)

¸

(Note: press

k

and select when, or simply type it.)

Observe the appearance of ‘else’ in the display. It means ‘for all other values of x’ (x≥0 in this case).

f x x x

x x

( )= − <

+ ≥ ⎧

⎨ ⎩

0 1 0

To evaluate the function, use the table feature: Press

¥

[TblSet]

Type tblStart = –5 Type

Δ

tbl = 5

Press

¸

to save the new values. Press

¥

[TABLE]

The table values required are displayed.

b Press

¥

[Y=]

Press

„

(Zoom), select 4:ZoomDec

The two sections of the graph appear and can be traced or evaluated in the usual ways.

4.6

How to work with hybrid functions

Some functions have different rules for different sections of their domain. They are sometimes called hybridfunctions or piecewise definedfunctions. The

when

function can be used to define and plot the graphs of hybrid functions.

E

E x a m p l e 1

The function fis defined by

a Enter the hybrid rule and evaluate the function for x= –5, 0, 5, 10.

b Plot the graph of the function.

Solution

a Press

¥

[Y=]

Type y1=when(x<0,–x,x^2+1)

¸

(Note: press

k

and select when, or simply type it.)

Observe the appearance of ‘else’ in the display. It means ‘for all other values of x’ (x≥0 in this case).

To evaluate the function, use the table feature: Press

¥

[TblSet]

Type tblStart = –5 Type

Δ

tbl = 5

Press

¸

to save the new values. Press

¥

[TABLE]

The table values required are displayed.

f x x x

x x

( )= − <

+ ≥ ⎧

⎨ ⎩

0 1 0

2

With hybrid functions, multiple evaluations are best done by substituting individual values at the home screen, or by using the table feature. Spurious results might be obtained if you try to evaluate the function at the home screen using the {…} method.

J

b Press

¥

[Y=]

Press

„

(Zoom), select 4:ZoomDec

and press

¸

The two sections of the graph appear and can be traced or evaluated in the usual ways.

E

E x a m p l e 2

The function fis defined by

a Evaluate the function starting at x= – /4, in increments of /4.

b Plot the graph of the function.

Solution

a Press

¥

[Y=]

Type y1=when(x<0 or x>

2

[

π

] ,1,

2

[SIN](x))

¸

The function is displayed in y1 in the above form. (Interpret ‘else’ to mean for all other values of x: 0 ≤x≤ in this case.)

To evaluate the function, use the table feature: Press

¥

[TblSet]

Type tblStart = –

2

[

π

] /4

Type

Δ

tbl =

2

[

π

] /4

Press

¸

to save the new values. Press

¥

[TABLE]

b Press

¥

[WINDOW]

and type the following values. (The multiples of are to provide good tracing of the sin section.)

xmin = –

2

[

π

] ÷4

xmax = 5

2

[

π

] ÷4

xscl =

2

[

π

] ÷4

ymin = –0.5

ymax = 1.5

yscl = 1

xres = 1

Press

¥

[GRAPH]

π π

π π

f x

x

x x

x

( ) = sin

<

≤ ≤

>

⎧ ⎨ ⎪

⎩ ⎪

1 0 0 1

4.7

How to work with reciprocal

functions

If fis a function with rule f(x), the related function with rule is called

the reciprocal function of f(in the same way as is the reciprocal of the number 5). To see the relationships between the two functions, it is useful to look at their graphs drawn on the same set of axes. For the special case of the reciprocal function with rule y= , the graph is a rectangular hyperbola.

E

E x a m p l e 1

For the function with rule y= x– 2, plot the graph of the function and its reciprocal function.

Solution

Press

¥

[Y=]

Type y1=x–2

¸

Type y2=1÷y1(x)

¸

Move the cursor back to y1.

Press

2

[F6] (Style), select 4:Thick

and press

¸

This gives a heavy line for ease of comparison.

Press

„

(Zoom), select 4:ZoomDec

and press

¸

The thick graph is the straight line, and the two curves represent the graph of the reciprocal function with rule y= .

To get a correct representation: Press

¥

[WINDOW]

Change xres to 1. Press

¥

[GRAPH]

Now we see a truer representation with no vertical line. Tracing shows why: at

x= 2, yis undefined (as 1/0 is undefined).

1 2

x−

1

x

1 5

1

f x( ) 1

f

Why the vertical line? Most CAS graphing software joins each plotted point to the next. In the above example, the line joins the last point plotted on the left curve, (1.9, –10), to the next point plotted, which in this case is (2.1, 10) on the right curve. (Check by tracing y2.) The vertical line is spurious: it does not belong to the graph. In the example above, selecting ZoomDecand setting xresto 1 removed this line.

J

E

E x a m p l e 2

Plot the graph of the reciprocal function with rule y= .

a On the same axes, plot the graph obtained by translating the reciprocal function 3 units to the right and 1 unit down.

b Express the rule of the transformed function in the form of a fraction with a common denominator.

Solution

a Press

¥

[Y=]

Type y1=1÷x

¸

Press

2

[F6] (Style), select 4:Thick

and press

¸

Press

„

(Zoom), select 4:ZoomDec

and press

¸

The transformed function has rule y= – 1. Press

¥

[Y=]

Type y2=1÷(x–3)–1

¸

Press

¥

[WINDOW]

Change xres to 1. Press

¥

[GRAPH]

As the initial graph has the x-axis (y= 0) and the y-axis (x= 0) as asymptotes, the transformed graph has asymptotes y= –1 and x= 3.

b Press

t

Press

„

(Algebra), select 6:comDenom(

and press

¸

Type comDenom(y2(x))

¸

The rule is equivalent to y= 4

3

− −

x x

1 3

x −

1

x

Another way to prevent spurious lines being drawn near vertical asymptotes is to change the line style in the Y=editor. To do this, place the cursor next to the relevant function, press ˆ, select 2:Dotand press ¸. Points that have been calculated will now not be joined.

Note that when the graph is steep, the spacing between calculated points may be unsatisfactory.

J

4.8

How to work with rational

functions

If fand gare polynomial functions, the related function is called a rational function(in the same way as is a rational number). Example 2 in section 4.7

shows that if fand gare both linear functions, then the rational function is simply a rectangular hyperbola (a dilation and translation of the standard hyperbola). In this section, we consider cases where one or both of fand gare quadratic.

E

E x a m p l e 1

For the rational function with rule y= :

a plot the graph of the function

b explain its asymptotic behaviour

c draw its asymptotes.

Solution

a Press

¥

[Y=]

Type y1=(x^2–x

−

2)÷(x+2)

¸

Press

„

(Zoom), select 8:ZoomInt

and press

¸

The graph seems almost linear but with a kink near the origin. What is going on? Like reciprocal functions, it can be difficult to get a good view.

Press

„

(Zoom), select 4:ZoomDec

and press

¸

We now have a completely different view: it looks nothing like the previous one!

Press

„

(Zoom), select 6:ZoomStd

and press

¸

An extra piece of the puzzle appears near the bottom of the screen.

x x

x

2 ₂

2

− −

+

2 5

f g

With rational functions, it is often useful to perform some algebraic manipulation of the function rules to show alternative forms. This can give some insight into the

behaviour of their graphs. A useful feature in this context (available from the home screen) is the

3:expand(

command on the

„

(Algebra)

menu.

J

b Clearly, there is a problem at x= –2 where the denominator is zero. As with reciprocal graphs, there will be a vertical asymptote there. To get a better view, press

„

(Zoom), select 4:ZoomDec

and press

¸

Press

¥

[WINDOW]

and enter the following values:

ymin = –20

ymax = 10

xres = 1

Press

¥

[GRAPH]

…

(Trace)

The spurious line is gone and we see a graph with branches on either side of the vertical asymptote. Also observe that the graph tends to become more linear as we move away from the asymptote.

Press

t

Press

„

(Algebra), select 7:propFrac(

and press

¸

Type propFrac(y1(x))

¸

or expand(y1(x))

¸

We have a combination of a hyperbola and a straight line. For large values of x, the first term is almost zero, so the graph is almost linear. The line

y=x– 3 is an obliqueasymptote.

c To draw the asymptotes on the graph, plot the linear rule. Press [Y=]

Type y2=x–3

¸

Press

¥

[GRAPH]

Press

2

[F7] (Pen), select 5:Vertical

and press

¸

Press

A

to x = –2 and press

¸

The graph now appears with its asymptotes.

E

E x a m p l e 2

For the rational function with rule y= :

a plot the graph of the function

b comment on its asymptotic behaviour.

Solution

a Press

¥

[Y=]

Type y1=(x+2)÷(x^2–x–2)

¸

Press

„

(Zoom), select 4:ZoomDec

and press

¸

There appear to be three sections, suggesting that there are two vertical asymptotes.

x

x x

+

− −

2 2

b Press

t

Press

„

(Algebra), select 3:expand(

and press

¸

Type expand(y1(x))

¸

The rule is equivalent to y= , so there are vertical

asymptotes at x= –1 and x= 2. For large values of x, both terms approach zero, so y= 0 is a horizontal asymptote.

Changing xresto 1 in the window settings will confirm the presence of two vertical asymptotes.

Press

¥

[WINDOW]

and set xresequal to 1. Press

¥

[GRAPH]

4.9

How to work with sum and

difference functions

If fand gare functions, the related function f+ gis called the sum function, and

f – gis called the difference function. Function graphing software makes it easy to plot the graph of a sum (or difference) function directly, but it can be useful to plot the graphs of fand gtoo. For example in a sum function, any x-value, the y-value or ordinate is given by y = f(x) + g(x), that is, the sum of the ordinatesof fand g. So, we can think of the graphing process in this case as

addition of ordinates. A similar process can be used for a difference function.

E

E x a m p l e

Plot the graphs of f, gand f+ gon domain Rwhere fand gare given by:

a f(x) = x, g(x) = cos(x)

b f(x) = sin(x), g(x) = cos(x)

Solution

a Make sure you are in radian angle mode. Press

¥

[Y=]

Type

y1=x

¸

Type

y2=

2

[COS] (x)

¸

Type y3=y1(x)+y2(x)

¸

Scroll to y3, press

ˆ

(Style), select 4:Thick

and press

¸

Press

„

(Zoom), select 6:ZoomStd

and press

¸

The thick curve shows the graph of y= x+ cos(x). Features to note include:

n when cos(x) = 0, the graph intersects with that of y= x(as we are adding 0 at these points)

4 3 2

1 3 1 / /

n when cos(x) >0, the graph lies above that of y= x(as we are adding a positive amount at these points)

n when cos(x) <0, the graph lies below that of y= x(as we are adding a negative amount at these points).

b Press

¥

[Y=]

Type y1=

2

[SIN] (x)

¸

Type y2=

2

[COS] (x)

¸

Press

„

(Zoom), select 7:ZoomTrig

and press

¸

The thick curve (y3) shows the graph of y= sin(x) + cos(x). The same key features as above hold. In particular, when cos(x) = 0, the graph intersects that of y= sin(x), and when sin(x) = 0, the graph intersects that of y= cos(x).

Turning off the plots of y1 and y2 and regraphing displays the graph of

y= sin(x) + cos(x) alone. The curve appears to be a dilated and translated sin (cos) curve.

4.10

How to work with products of

functions

If fand gare functions, the related function f

×

gis called the product function. Using a CAS makes it easy to plot the graph of a product function directly, but as with addition of ordinates, it can be useful to plot the graphs of fand gas well. For any x-value, the y-value or ordinateis given by y= f(x)

×

g(x), that is, the product of the ordinatesof f and g. Key points on the product function graph occur when either function is equal to 0 or ±1.

E

E x a m p l e

Plot the graphs of f, gand f

×

gon domain Rwhere fand gare given by:

a f(x) = x, g(x) = cos(x) b f(x) = sin(x), g(x) = –e0.2x

It is a useful practice to try to sketch the shape of the sum curve by first sketching the shapes of fand g, then using the key ideas shown in the example above. You can then use the CAS to plot the graph and check your sketch.

J

J

C O M M E N T

C O M M E N T

In the above example, turning off the plots of y1 and y2 (by pressing

†

in the Y= editor) and regraphing displays the graph of y= x+ cos(x) alone.

J

Solution

a Make sure you are in radian angle mode. Press

¥

[Y=]

Type y1=x

¸

Type y2=

2

[COS] (x)

¸

Type y3=y1(x)

y2(x)

¸

Scroll to y3, press

2

[F6] (Style), select 4:Thick

and press

¸

Press

„

(Zoom), select 6:ZoomStd

and press

¸

The thick curve shows the graph of y= xcos(x). Features to note include:

n when xor cos(x) is zero, the graph crosses the x-axis (as we are multiplying by 0 at these points)

n when cos(x) = 1, the graph touches that of y= x(as we are multiplying

xby 1 at these points)

n when cos(x) = –1, the graph touches that of y= –x(as we are

multiplying xby –1 at these points). Check this feature by plotting the graph of y= –x.

Press

¥

[Y=]

Type y4=–x

¸

Press

¥

[GRAPH]

Turn off the plot of y2 (by pressing

†

in the Y= editor) and regraph to better see this.

This view shows that the graph is sandwiched between the graphs of y= x

and y= –x. (Why is this?)

You can also turn off the plots of y1 and y4 to show the graph of

y= xcos(x) alone.

b Press

¥

[Y=]

Type

y1=

2

[SIN] (x)

¸

Type y2=e^(0.2x)

¸

Type y3=y1(x)

y2(x)

¸

Press

„

(Zoom), select 4:ZoomDec

and press

¸

The thick curve shows the graph of y= –sin(x)e0.2x. The same key features as above hold:

n when sin(x) = 0 the graph crosses the x-axis

n when sin(x) = 1 the graph touches that of y= –e0.2x

n when sin(x) = –1, the graph touches that of y= e0.2x.

Adding the graph of y= e0.2xand turning off y1 shows that the graph is again sandwiched between the graphs of y = −e0.2xand y= e0.2x.

Turning off all plots except y3 and regraphing displays the graph of

y= sin(x)e0.2xalone.

4.11

How to work with composite

functions

If fand gare functions, the related functions with rules f(g(x)) and g(f(x)) are called composite functions. It follows that the maximal domain for a composite function is at most the domain of the inner function. However, as the inner function is evaluated first, its value is substituted into the outer function, and this can only work if this value is an element of the domain of the outer function. So, the range of the inner function must be a subset of the domain of the outer function for the composition to be defined. Hence, the implied domain of a composite function is the domain of the inner function or a subset of this domain.

It is a useful practice to try to sketch the shape of the product curve by first sketching the shapes of fand g, then using the key ideas shown in the example above. You can then use the CAS to plot the graph and check your sketch.

J

J

C O M M E N T

C O M M E N T

CAS software makes it easy to plot the graph of the ruleof a composite function directly, but care must be taken with domain restrictions. Your calculator can perform substitution and simplification of rules but it cannot handle domain restrictions. You must determine the necessary restrictions yourself.

J

E

E x a m p l e

For the functions fand gwith rules below, state their implied domains. Also give the maximal implied domains of the composition functions with rules

f(g(x)) and g(f(x)), state their rules in simplified form, and plot the graphs of

y= f(g(x)) and y= g(f(x)).

a f(x) = x3, g(x) = cos(x) b f(x) = x2, g(x) =

Solution

a The implied domain of each function is R, so each composition is also defined over domain R.

Press

t

Press

†

(Other), select Define

and press

¸

Type Define f(x)=x^3

¸

Type Define g(x)=

2

[cos] (x)

¸

Type f(g(x))

¸

Type g(f(x))

¸

The first composition rule is f(g(x)) = (cos(x))3= cos3(x). Press

¥

[Y=]

Type y1=f(g(x))

¸

Press

„

(Zoom), select 7:ZoomTrig

and press

¸

The graph intersects the x-axis at the same places as the graph of

y= cos(x), as expected. Maximum and minimum points are also at the expected places.

The second composition rule is g(f(x)) = cos(x3). Press

¥

[Y=]

Alter the rule to y1=g(f(x))and press

¸

Press

¥

[GRAPH]

This looks weird! The difficulty is due to the fact that intercepts become more and more frequent (when x3= π/2, 3π/2, 5π/2 etc.) than for cos(x), and the limited number of pixels on the calculator screen makes it difficult to get a good plot.

Press

¥

[WINDOW]

and enter the following values:

xmin = –3

xmax = 3

xscl = 1

ymin = –1.5

ymax = 1.5

xres = 1

Press

¥

[GRAPH]

We get a better view, showing that the effect of the inner function is to compress the graph of y= cos(x).

b The implied domain of fis Rand the implied domain of gis [0, ∞). The

range of fand gare both [0, ∞) (check their graphs). Then ran g⊆dom f, so the composition f(g(x)) is defined on domain [0, ∞) = dom g. Similarly, ran f⊆dom g, so the composition g(f(x)) is defined on R= dom f.

Press

t

Press

†

(Other), select Define

and press

¸

Type Define f(x)=x^2

¸

Type Define g(x)=

2

[

√

] (x)

¸

Type f(g(x))

¸

Type g(f(x))

¸

The first composition rule is f(g(x)) = x. Press

¥

[Y=]

Type y1=f(g(x))

Í

x

≥

0

¸

Press

„

(Zoom), select 4:ZoomDec

and press

¸

The domain restriction is essential! If it is omitted, the graph of y= xwill be plotted for all real values of x, but the composite function is only defined for [0, ∞).

The second composition rule is g(f(x)) = |x|. Press

¥

[Y=]

Type y1=g(f(x))

¸

Press

¥

[GRAPH]

As the domain here is R, no restriction is necessary.

4.12

How to work with functional

equations

Functional equations are equations involving unknown functions that have to be found. Two examples are f(kx) = kf(x), where k is a constant, and

f(x+ y) = f(x) + f(y). The problem is often to find a function typethat satisfies the equation. Indeed, there may be many different classes of function that solve it. One of the main reasons for looking at functional equations is to facilitate a better understanding of functional behaviour.

E

E x a m p l e 1

Consider the functional equation f(2x) = 2f(x).

a Show that the circular function sinedoes not satisfy this functional equation.

b Show that the circular function cosinedoes not satisfy this functional equation.

Using the

Define

command (on the †menu) is a particularly useful feature. Function composition and algebraic simplification are powerful CAS tools considered in this topic.

J

c Determine which, if any, of the following function types satisfy this functional equation:

(i) a linear function whose graph passes through the origin

(ii) a quadratic function whose graph has its vertex at the origin

(iii) the exponential function with rule f(x) = ex.

Solution

a Press

t

Press

†

(Other), select 1:Define

press

¸

Type Define f(x)=

2

[sin] (x)

¸

Type f(2x)

¸

Type 2

f(x)

¸

The two expressions do not appear to be identical, but perhaps the first simplifies to the second. Check this using some algebra.

Press

„

(Algebra), select 9:Trig

and 1:tExpand(and press

¸

Type

tExpand(

2

[sin] (2x))

¸

It is now clear that the two expressions are not identical, so sinedoes not satisfy the equation f(2x) = 2f(x).

b Repeat with the cosinefunction.

Press

†

(Other), select 1:Define

press

¸

Type Define f(x)=

2

[cos] (x)

¸

Type f(2x)

¸

Type 2

f(x)

¸

Press

„

(Algebra), select 9:Trig

and 1:tExpand(and press

¸

Type tExpand(

2

[cos] (2x))

¸

It is again clear that the two expressions are not identical, so cosinedoes not satisfy the equation f(2x) = 2f(x).

c (i) To define a linear function with a graph through (0, 0):

Type Define f(x)=m

x

¸

Type f(2x)

¸

Type 2f(x)

¸

The two expressions are identical, so linear functions of this type satisfy the equation f(2x) = 2f(x). What does this mean?

(ii) To define a quadratic whose graph has vertex (0, 0): Type Define f(x)=a

x^2

¸

Type f(2x)

¸

Type 2f(x)

¸

(iii) To define the exponential:

Type Define f(x)=e^(x)

¸

Type f(2x)

¸

Type 2f(x)

¸

The two expressions are clearly not identical (for example, if x= 0, the first is 1 and the second is 2) so the exponential does not satisfy the equation f(2x) = 2f(x).

E

E x a m p l e 2

Consider the functional equation f(x + y) = f(x)f(y). Which, if any, of the following function types satisfy this functional equation?

a The square root function with rule f(x) = .

b The cos function with rule f(x) = cos(x).

c The exponential function with rule f(x) = ex.

Solution

a Press

t

Press

†

(Other), select 1:Define

and press

¸

Type Define f(x)=

2

[

√

] (x)

¸

Enter f(x+y)

¸

Enter f(x)

f(y)

¸

The two expressions do not appear to be identical. The easiest way to check this is to try an evaluation. It is easy to see that if x= 1 and y= 1, then f(x+ y) =√(1 + 1) = √2 whereas f(1) ×f(1) = √1 ×√1 = 1. So, the two

expressions are not identical and the square root function does not satisfy the equation f(x+ y) = f(x)f(y).

b Repeat with the cosine function:

Type Define f(x)=

2

[cos] (x)

¸

Type f(x+y)

¸

Type f(x)

f(y)

¸

The two expressions do not appear to be identical. We can check with an evaluation or with some algebra.

Press

„

(Algebra), select 9:Trig

and 1:tExpand(and press

¸

Type tExpand(

2

[cos] (x+y))

¸

It is clear that the two expressions are not identical and the cos function does not satisfy the equation f(x+ y) = f(x)f(y).

c Repeat with the exponential function:

Type Define f(x)=e^(x)

¸

Type f(x+y)

¸

Type f(x)

f(y)

¸

Calculus

5.1

How to calculate average and

instantaneous rates of change

The average rate of change for a function on [a, b] is .

Alternatively, it can be calculated using where his an increment added to a given xvalue. The easiest way to do this with your CAS is to define your function, , and then use this notation in subsequent calculation.

E

E x a m p l e

a Find the average rate of change for the function between the following values:

(i) x = 2 and x = 2.1 (ii) x = 2 and x = 2.05

(iii) x = 2 and x = 2.001 (iv) x = 2 and x = 2 + h

b Use limits and the above results to find the instantaneous rate of change at x= 2.

Solution

a Press

†

(Other), select 1:Define

Type Define f(x)=x^2

¸

You can now just