*Inverse Relations

*Objectives: Students will be able to…

*Determine whether a function has an inverse

*Write an inverse function

*Verify 2 functions are inverses of each other

3

x f(x)

33333 9999

999

y f-

1(x)

9999999 3333

333

x2 √ x

* The inverse of a given function will “undo” what the original function did.

* For example, let’s take a look at the square function: f(x) = x2

*Inverse Relations:

*The inverse of f is f -1 (read “f inverse”)

*If both the original relation and the inverse relation are both functions, they are inverse functions!

*The domain of the original relation is the range of the inverse.

Graphically, the x and y values of a function and its inverse are switched.

If the function y = g(x) contains the points

then its inverse, y = g-

1(x), contains the points

x 0 1 2 3 4

y 1 2 4 8 16

x 1 2 4 8 16

y 0 1 2 3 4

Where is there a line of reflection?

The graph of a

function and its

inverse are mirror images

about the line

y = x

y = f(x)

y = f-

1(x)

y = x

*Find the inverse relation:

x -3 -1 5 7 10

y -5 -3 0 4 7

x

y

Writing the equation of an inverse function:

Example 1: y = 6x - 12

Step 1: Switch x and y: x = 6y - 12

Step 2: Solve for y: x 6y 12x 12 6yx 126

y

1

6x 2 y

*Find an equation for the inverse relation:

1. 2. 712 xy7

5

7

3 xy

*Verifying 2 functions are inverses

* If f and g are inverse functions, their composition would simply give x back.

xxgfgf xxfgfg

*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.

*Because f(g(x))=x and g(f(x))=x, 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

2;2,2 2 xxgxxxf

xxxgf 22

22

xxxfg 2222 2

Since both of these = x, if you start with x and apply the functions they “undo” each other and are inverses.

*Verify that the functions f and g are inverses of each other.

*Remember…

The graph of a function needs to pass the vertical line test in order to be a function.

*Horizontal Line Test

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

*If the original function passes the horizontal line test, then its inverse will pass the vertical line test and therefore is a function.

*If the original function does not pass the horizontal line test, then its inverse is not a function (may need to restrict domain so that it has an inverse).

*One to One Functions

*All functions have only one y value for any given x value (this means they pass the vertical line test)

*One-to-one functions also have only one x-value for any given y-value.

If a horizontal line intersects the graph of an equation more than one time, the equation graphed is NOT a one-to-one function and will NOT have an inverse function.

This is a one-to-one function

This is NOT a one-to-one function

This is NOT a one-to-one 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.

*But what if I restricted the domain of x2? Then it would

have an inverse. Graph y = x2 for x > 0. Find the inverse function.

*Ex: Graph f(x)=2x2-4. Determine whether f -1(x) is a function.

*How could I restrict the domain to make it have an inverse function?

f -1(x) is not a function.

x > 0

Example 2:

Given the function : y = 2x2 -4, x > 0 find the inverse

Step 1: Switch x and y:x = 2y2 -4

Step 2: Solve for y:

yx

yx

yx

yx

2

4

2

4

24

42

2

2

2

Only need the positive root for inverse!

*Ex: Write the inverse of g(x)=2x3

Inverse is a function!

y=2x3

x=2y3

3

2y

x

yx

3

2

3

2

xy

2

43 xy

*Graph the functions to determine whether their

inverses will also be functions