SOLUTIONS. Would you like to know the solutions of some of the exercises?? Here you are

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SOLUTIONS

Would you like to know the solutions of some of the exercises?? Here you are… Your first “function”: remember

Introduction:

Text:

1)

b) It is not a function because for every value of X (different of zero) there are TWO corresponding value for Y, NOT only ONE.

c) Corrected wrong words: algebraic, function, expression, usually, example.

2)

Function 1:

The algebraic expression of the function is y = 3x Function 2:

The algebraic expression of the function is y = x/3

A  function  is  a  relationship  between  two  sets  by  which  we  assign  to  each  element  of   the  first  set  one  element  of  the  second.  

Usually  the  sets  are  numerical  (magnitudes)  and  their  elements  are  values.  The  first   one  is  the  independent  variable  (it’s  common  to  identify  it  with  the  letter  X)  and  the   second  one  is  the  dependent  variable  (usually  identified  with  the  letter  Y).  

X=Nº of candies you buy 1 2 3

Y= Price you pay (€) 2 4 6 18

x 3 9 30 45

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Function 3:

The algebraic expression of the function is y = x+20 Function 4:

The algebraic expression of the function is y = x3 Function 5:

The algebraic expression of the function is y = +√x

4) The sentences are:

• A function is a relationship between two magnitudes.

• The algebraic representation of a function shows the relationship by a formula.

• The graphic representation of a function uses a graphic on the Cartesian plane to show the relationship.

Work shit: 1)

e) Ana thinks that a function is a relationship between two sets, but she forgets an important condition: for every element of the first set you assign only of the second.

f) When you write in two rows (or two columns) the values of the two magnitudes that are related by a function, you are creating a value table.

g) When you draw on the Cartesian plane points obtained from a table value (the first coordinate is the value for the independent variable and the second coordinate is the corresponding value for the dependent variable, calculated using the algebraic expression) you do the graphic representation of the function.

x 1 2 7 40 y 21 22 27 60 x 1 2 3 4 y 1 8 27 64 x 4 9 16 225 y 2 3 4 15

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3)

Only the first one is really a function. In the other cases there is more than one possibility for the dependent value.

4)

Graphic representation of a function:

Value for the

independent variable X dependent variable YValue for de the Algebraic expression of the function Y=f(x)

3 6 y=2x or y=x+3

4 16 y=4x, y=x2 _{or y=x+12}

6 2 Y=x/3

11 5 Y=x-6

100 10 y=10x, y=√x y=x-90

150 1,5 y=x/10 0 3 Y=3+x5 1 4 Y=3+√x  

Y=  3x+2  

Y=  2  

Y=  3x

+2  

Y=  -­‐x

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1)

a) No, it’s not possible to divide by zero.

b) No, to know the value of f(1) you need to calculate the square root of a negative number.

c) Is not defined in certain cases.

3) Range of a function 1) a) 16, 9, 16, 9. b) Positive. c) Positive.

2) The values of Y are always 0 or a whole number.

3) The set of possible values for Y produced by the function. 4)

b)

The  domain  of  a  function  y=f(x)  is  the  set  of  values  for  which   the  function  is  defined,  usually  denoted  Domf(x).  

Text:    

The   range   of   a   function   y=f(x)   is   the   set   of   all   values   of   the   dependent   variable   y   produced   by   f   from   all   the   elements   of   the   domain    

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Increasing, decreasing and more 5)

6) Sentences:

• Y = f(x) is a increasing function if when a b then f(a) f(b) for every values a and b of Domf(x). • Y = f(x) is a decreasing function if when a b then f(a) f(b) for every values a and b of Domf(x). • The largest value of the range of a function is the maximum of the function.

• The smallest value is the minimum value of the function. If we consider the whole function these points are global maximum or global minimum.

Continuous functions

4) Continuous, graph, small, value, independent, variable. Linear functions

2)

(*) it fits every function!!

Local  minimum   Global  minimum   Local  maximum   Global  maximum   (-­6,-­3.9)   (6,3.3)   (1,2)   (-­2,-­1.5)     Sentence Function Is a decreasing function Y = 6X Is an increasing function Y = 0.5X

(0,0) is a point of the graphic (*) Y = -0.25X

Has every value of X on the domain (*) Y=-0

Is a constant function Y=-7x

If X increases one unit, Y increases six

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Text:

The number m is called the slope of the function, and it means the number of units that the dependent variable increases (or decreases) for every unit that the independent variable increases (or decreases).

The point (0,0) is always on the graphic of a linear function.

If the slope is positive, the function is increasing. If the slope is negative, the function is decreasing. The function is constant if the slope equals zero.

In the situation of the association the domain of the function wasn’t the set of all the numbers, but in general the domain of a linear function includes all numbers.