7.4 Inverse Functions7.4 Inverse Functions

p. 422p. 422

Review from chapter 2Review from chapter 2

• Relation – a mapping of input values (x-values) onto output values (y-values).

• Here are 3 ways to show the same relation.

y = x2 x y

-2 4

-1 1

0 0

1 1

Equation

Table of values

Graph

• Inverse relation – just think: switch the x & y-values.

x = y2

xy

x y

4 -2

5 -1

0 0

1 1

** the inverse of an

equation: switch the x

& y and solve for y. ** the

inverse of a table:

switch the x & y.

** the inverse of a graph: the reflection of the original graph

in the line y = x.

Ex: Find an inverse of y = -3x+6.• Steps: -switch x & y

-solve for y

y = -3x+6

x = -3y+6

x-6 = -3y

yx

3

6

23

1

xy

Inverse Functions

• Given 2 functions, f(x) & g(x), if f(g(x))=x AND g(f(x))=x, then f(x) & g(x) are inverses of each other.

Symbols: f -1(x) means “f inverse of x”

Ex: Verify that f(x)=-3x+6 and g(x)=-1/3x+2 are inverses.

• Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses.

f(g(x))= -3(-1/3x+2)+6

= x-6+6

= x

g(f(x))= -1/3(-3x+6)+2

= x-2+2

= x

** Because f(g(x))=x and g(f(x))=x, ** Because f(g(x))=x and g(f(x))=x, they are inversesthey are inverses..

To find the inverse of a function:To find the inverse of a function:

1. Change the f(x) to a y.

2. Switch the x & y values.

3. Solve the new equation for y.

** Remember functions have to pass the vertical line test!

Ex: (a)Find the inverse of f(x)=x5.

1. y = x5

2. x = y5

3. 5 55 yx

yx 5

5 xy

(b) Is f -1(x) a function?

(hint: look at the graph!

Does it pass the vertical line test?)

Yes , f -1(x) is a function.

Horizontal Line TestHorizontal Line Test

• Used to determine whether a function’s inverseinverse will be a function by seeing if the original function passes the horizontal horizontal line testline test.

• If the original function passespasses the horizontal line test, then its inverse is a inverse is a functionfunction.

• If the original function does not passdoes not pass the horizontal line test, then its inverse is not inverse is not a functiona function.

Ex: Graph the function f(x)=x2 and determine whether its inverse is a

function.

Graph does not pass the horizontal line test, therefore the inverse is not a function.

Ex: f(x)=2x2-4 Determine whether f -1(x) is a function, then find the inverse equation.

2

2

4y

x

f -1(x) is not a function.

y = 2x2-4

x = 2y2-4

x+4 = 2y2

2

4x

y

22

1 xyOR, if you fix the

tent in the basement…

Ex: g(x)=2x3

Inverse is a function!

y=2x3

x=2y3

3

2y

x

yx

3

2

3

2

xy

OR, if you fix the tent in the basement…

2

43 xy

Assignment