Inverse Functions and their Representations

Lesson 5.2

Definition

A function is a set of ordered pairs with no two first elements alike. f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) }

But ... what if we reverse the order of the pairs? This is also a function ... it is the inverse function f -1(x) = { (x,y) : (2, 3), (4, 1), (6, 7), (12, 9) }

Example

Consider an element of an electrical circuit which increases its resistance as a function of temperature.

T = Temp R = Resistance

-20 50

0 150

20 250

40 350

R = f(T)

Example

We could also take the view that we wish to determine T, temperature as a function of R, resistance.

R = Resistance T = Temp

50 -20

150 0

250 20

350 40

T = g(R)

Now we would say that g(R) and f(T) are inverse

functions

Terminology

If R = f(T) ... resistance is a function of temperature,

Then T = f-1(R) ... temperature is the inverse function of resistance. f-1(R) is read "f-inverse of R“ is not an exponent it does not mean reciprocal 1 1

xx

Does This Have An Inverse? Given the function at the right

Can it have an inverse? Why or Why Not?

NO … when we reverse the ordered pairs, the result is Not a function We would say the function is

not one-to-one

A function is one-to-onewhen different inputs always result in different outputs

x Y

1 5

2 9

4 6

7 5( ) ( )c d f c f d

One-to-One Functions

When different inputs produce the same output Then an inverse of the function does not exist

When different inputs produce different outputs Then the function is said to be “one-to-one”

Every one-to-one function has an inverse Contrast

( ) ( )c d f c f d

2 3( ) and ( )f x x g x x

One-to-One Functions

Examples

Horizontal line test?

3( )f x x

2( )f x x

Finding the Inverse

Try2

2

xy

x

1

Given ( ) 2 7

then 2 7

7solve for x x

27

2

f x x

y x

y

yf y

Composition of Inverse Functions

Consider f(3) = 27   and   f -1(27) = 3 Thus, f(f -1(27)) = 27 and f -1(f(3)) = 3

In general   f(f -1(n)) = n   and f -1(f(n)) = n

(assuming both f and f -1 are defined for n)

3 1 3( ) ( )f x x and f x x

Graphs of Inverses

Again, consider Set your calculator for the functions shown

3 1 3( ) ( )f x x and f x x

Dotted style

Use Standard Zoom

Then use Square Zoom

Graphs of Inverses

Note the two graphs are symmetric about the line y = x

Investigating Inverse Functions

Consider

Demonstrate that these are inverse functions What happens with   f(g(x))? What happens with  g(f(x))?

3

3

( ) 2 4

( ) 48

f x x

xg x

Define these functions on your calculator and try

them out

Domain and Range

The domain of f is the range of f -1 The range of f is the domain of f -1

Thus ... we may be required to restrict the

domain of f so that f -1 is a function

Domain and Range

Consider the function    h(x) = x2 - 9 Determine the inverse function

Problem =>  f -1(x) is not a function

Assignment

Lesson 5.2 Page 396 Exercises 1 – 93 EOO