Exponential Functions

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        





















y = 2 ^x y

x

Exponential Functions

Definition: An Exponential Function is an function that has the form ax

x

f ( )  , where a > 0. The number a is called the base.

Example:Let f (x)2^x It is clear what the function means for some values of x.

For example

1 2 ) 0

(  ⁰ 

f , f (1) 2¹  2 ,

4 2

) 2

(  ² 

f , f (3)  2³ 8,

2 2 1

) 1

(  ^¹ 

f ,

4 2 1

) 2

(  ^² 

f ,

414 . 1 2 2

)

(₂¹  ¹²  

f ,

and f(3.2)  2³^.²  2³ 2¹⁵  8⁵ 2 .

Defining ^f ⁽^x⁾^²^x for x irrational is too difficult now. This graph represents exponential growth since it is increasing as x increases.

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y

        

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

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

y = (1/2) ^x= 2^{- x} y = 2 ^x

x Recall that replacing x with –x in a function amounts to a rotation of the graph about the y-axis.

So we can get the graph of ^x

x

f   ^



 



  2

2 ) 1

( , from f (x)2^x_shown previously.

It is shown dotted below.

The function

x

f 



 



  2 ) 1

(

represents exponential decay since the function decreases as x increases.

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           x y





















Exponential Growth:

All functions f (x)  a^x where a > 1, exhibit exponential growth.

y  3

x

y  10

x

y  2

^x

y  1 . 5

x

y  1 . 1

x

y  4

x

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x

          

y























Exponential Decay:

All functions f (x) a^x where 0 < a < 1, exhibit exponential decay.

y  0.6x

y  0.4x

y  0.7x

y  0.8x

y  0.9x

 

^x

y  0.5x  ₂¹

 

^x

y  0.1x  ₁₀¹

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Alternate representation for exponential decay functions.

We can write ^x

 

^x ^x ^b ^x

a a a

x

f ^

 

  



 



 



 1

)

( ¹ , where

b  a1

So that f (x)  a^x where 0 < a < 1, can be written as f (x)  b^^x where b > 1.

Examples: ^x

x

y   ^



 



  10

10 1

x x

y x





 



 



 



 

 2

5 5

4 2 . 0

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“THE” Exponential Function e

^x

There is only one exponential function that has a slope of 1 at the point (0, 1).

This is called “the” exponential function and we denote the base with the letter e.

We will find out that e = 2.718281828459045

ex

x f ( ) 

-5 -4 -3 -2 -1 0 1 2 3 4 5 x

1 2 3 4 5 6 7 8 9

y

10

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x

-5 -4 -3 -2 -1 0 1 2 3 4 5

1 2 3 4 5 6 7 8

y

9 10

Important related functions are f (x)  e^k^x for real k.

If k > 0, f (x)  e^k^x

exponential growth functions.

If k < 0, f (x)  e^k^x

exponential decay functions.

e x

x

f ( )  ^

e x

x

f ( )  ⁰^.²⁵^* e x

x

f ( )  ²

e x

x

f ( )  ^⁰^.⁵^*

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Inverse Functions

Definition: a)We say f (x) and g(x) are inverse to each other if 1) f



g(x)



x

2) g



f (x)



x,

3) domain of f = range of g 4) domain of g = range of f.

b)The inverse of a function f is denoted f ^¹.

Domain of f Range of f

Range of g Domain of g

f

g

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Examples of Inverse Functions

Tabular functions: Just interchange the columns.

f (x) f ^¹(x)

x y

1 2

4 3

6 2 8 3

9 6

12 8

15 

16 1 18 5

x y

2 1

3 4

2 6 3 8

6 9

8 12

4 15

1 16 5 18

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(a, b) (b, a)

b a

a b

y = x

Finding the graph of an inverse function from the graph of the function.

If the point (a, b) is on the graph of f (x), then the point (b, a) is on the graph of )

1( x

f ^ . This represents a rotation of the graph of f (x) about the line y = x.

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x

-5 -4 -3 -2 -1 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4

y

5 y = x

Example:

The function f (x)  3x has the property that it triples the input, so it shouldn’t surprise you that the inverse function is one that would take one third of the input, or

) 3

1( x

x

f ^  . This is shown in the graph below.

x y  3

3 y  x

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x

-5 -4 -3 -2 -1 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4

y

5 y = x

Example:

Consider the function f (x)  x2 1. Thinking about the operations you first double then input then add one to the result. To get the inverse function undo the function, i.e. first subtract one then halve the result, or

2 ) 1

1(  

 x

x

f . This

is shown in the graph below.

1 2 

 x y

2

1

 x y

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When does a function have an Inverse?

For a function to have in inverse it must pass the horizontal line test. That is, any horizontal line must cross the graph of the function at most once.

If the two different points (a₁,b) and (a₂,b) are both on the graph of y  f (x), then the two points (b,a₁) and (b,a₂) would be on the graph of y  f ^¹(x). But this would mean that y  f ^¹(x) would have two different values in its range associated with the same value in its domain. This violates the definition of a function. This happens with the function f (x)  . x²

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x

-5 -4 -3 -2 -1 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4

y

5

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x

-5 -4 -3 -2 -1 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4

y

5

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Finding Inverse Functions given Expressions

When given an expression y  f (x), one may find an inverse function by interchanging x and y, then solving the resulting equation for y.

Example: Find the inverse of f (x)  x2 1.

So, y  x2 1

First interchange x and y. This gives x  y2 1. Now solve for y.

1 2 

 y x

y

x 1 2 or 2 y  x 1

2

1

 x

y so

2 ) 1

1(  

 x

x

f .

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Example: Find the inverse of f (x)  x³  4. So, y  x³ 4

First interchange x and y. This gives x  y³ 4. Now solve for y.

3 4

 y x

4 y3

x   or y³  x  4

3  4

 x

y so f ^¹(x)  ³ x  4.

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When can this procedure break down?

1) When you cannot solve for y.

Example: Let f (x)  x⁴  4x³  x

Then y  x⁴ 4 x³  x, and upon interchanging variables we get, x

y y

y⁴  4 ³   to be solved for y.

This is difficult if not impossible.

2) When an inverse doesn’t exist.

Example: Find the inverse of f (x)  x⁶3.

 Write y  x⁶3, and interchange variables to give x  y⁶ 3 y⁶  x3 or y  ⁶ x3

Aha, you see, that’s not a function. Of course, you knew that f ( x) does not have an inverse, because it flunks the horizontal line test.