1.8 Inverse functions My domain is your range No! My range is your domain.

1.8 Inverse functions

My domain is your rangeNo! My range is your domain

Input/Output table to show how this works

If g(x) = f -1(x) , then the range of f(x) is the domain of g(x).

x f(x) x g(x)1 9 9 12 8 8 2 3 7 7 3Can you think of the equations f(x) and

g(x)?

f(x) = - x + 10g(x) = - x + 10 To check to see if they are inverses we

should use composition of functions.(f º g)(x) = - (- x + 10 ) + 10

= x – 10 + 10 so (f º g)(x) = x

(g º f)(x) = - (- x + 10 ) + 10Same problem so (g º f)(x) = x

Lets try another set of inverse functions

f(x) = 3x + 2 g(x) = 1/3x – 2/3 (f º g)(x) = 3(1/3x – 2/3) + 2(f º g)(x) = x

(g º f)(x) = 1/3(3x + 2) – 2/3(g º f)(x) = x

What can you say about inverse functions?

To check if 2 functions are inverse

Use the compositions of the two functions,

namely f(x) and g(x).

(f º g)(x) = x(g º f)(x) = x

The value will always be x

Graphing inverse functions

h(x) = x2 and f(x) = √x

The inverses reflects over the line y = x (Identity function)

h(x) = x2 and f(x) = √x

This graph does not look complete why?

Horizontal Line Test

If you pass a horizontal line through the graph and it only touches in one point, then the function has an inverse.

Does not have an inverse, unless we limit the

domain

Horizontal Line Test

Does this function have an inverse?

One to One functions

For every dependent variable (output) there is only one independent variable

(input). Look at f(x) = 2x -3 If f(x) = 1, then x = 2If x = 2, then f(x) = 1

A function only has an inverse if and only if it is “one to one”.

Would f(x) = x2 have an inverse?

If f(x) = 16, then x = 4 or – 4.

Which makes f(x) not one to one, so no inverse.

How to find the Inverse

Given f(x) = 8x – 2, change f(x) to y y = 8x – 2

Interchange x and yx = 8y – 2

Solve for y. Add 2 and divide by 8

⅛x + ¼ = y ; so f -1(x) = ⅛x + ¼

How would you check to make sure the functions are inverses?

f(f -1) = 8(⅛x + ¼) – 2 = x + 2 – 2 = x

f-1(f) = ⅛(8x – 2) + ¼ = x - ¼ + ¼ = x

5 1)( xxf

5 1)( xxf

5 1 yx

5 1)( xxf

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5 1 yx

15 yx

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Would it change if y was y2 ?

5 1)( xxf

5 1 yx

15 yx

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1)( 51 xxf

Homework

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Homework

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