1. Inverse Relations and Functions
OBJ:  Find the inverse of a relation
 Draw the graph of a function
and its inverse
 Determine whether the
inverse of a function
is a function

2. FINDING INVERSES OF LINEAR FUNCTIONS
An inverse relation maps the output values back to their original input values. This means that the domain of the inverse relation is the range of the original relation and that the range of the inverse relation is the domain of the original relation. x 4 2
– 2
– 4 y 4 2
– 2
– 4 x 2 1
– 1
– 2 y 2 1
– 1
– 2
Original relation
Inverse relation
DOMAIN
DOMAIN
RANGE
RANGE

3. FINDING INVERSES OF LINEAR FUNCTIONSx y 4 2
– 2
– 4 1
– 1
Original relation
Inverse relation
– 2 4
– 1 2 1
– 2 2
– 4
Graph of original
relation
y = x
Reflection in y = x
Graph of inverse relation

4. FINDING INVERSES OF LINEAR FUNCTIONS
To find the inverse of a relation that is given by an equation in x and y, switch the roles of x and y and solve for y (if possible).

5. Finding an Inverse Relation
Find an equation for the inverse of the relation y = 2 x – 4.
SOLUTION
y = 2 x – 4
Write original relation.
x = 2 y – 4 x y
Switch x and y .
x + 4 = 2 y 4
Add 4 to each side.
x + 2 = y 1 2 2
Divide each side by 2.
The inverse relation is y = x + 2. 1 2
If both the original relation and the inverse relation happen to be functions, the two functions are called inverse functions.

6. f (g (x)) = x and g ( f (x)) = x
Finding an Inverse Relation
I N V E R S E F U N C T I O N S
Functions f and g are inverses of each other provided:
f (g (x)) = x and g ( f (x)) = x
The function g is denoted by f – 1, read as “f inverse.”
Given any function, you can always find its inverse relation
by switching x and y.
For a linear function f (x ) = mx + b where m  0, the
inverse is itself a linear function.

7. ( ) f (g (x)) = f x + 2 1 2 Show that f (g (x)) = x and g (f (x)) = x.
Verifying Inverse Functions
Verify that f (x) = 2 x – 4 and g (x) = x + 2 are inverses. 1 2
SOLUTION
Show that f (g (x)) = x and g (f (x)) = x.
f (g (x)) = f x + 2 1 2 ( )
= x – 4
= x + 4 – 4
= x
g (f (x)) = g (2x – 4)
= (2x – 4) + 2
= x – 2 + 2
= x 1 2

8. f (x) = x 2 Find the inverse of the function f (x) = x 2. y = x 2
FINDING INVERSES OF NONLINEAR FUNCTIONS
Finding an Inverse Power Function
x  0
Find the inverse of the function f (x) = x 2.
SOLUTION
f (x) = x 2
Write original function.
y = x 2
Replace original f (x) with y.
x = y 2
Switch x and y.
± x = y
Take square roots of each side.

9. f (x) = x 2 f (x ) = x 2 FINDING INVERSES OF NONLINEAR FUNCTIONS
The graphs of the power functions f (x) = x and g (x) = x 3 are shown along with their reflections in the line y = x.
f (x) = x 2
g (x) = x 3
On the other hand, the graph of g (x) = x 3 cannot be intersected twice with a horizontal line and its inverse is a function.
Notice that the graph of f (x) = x 2 can be intersected twice with a horizontal line and that its inverse is not a function.
Notice that the inverse of g (x) = x is a function, but that the inverse of f (x) = x is not a function.
inverse of g (x) = x 3
inverse of f (x) = x 2
g (x ) = x 3
g –1(x ) = x 3
f (x ) = x 2
x = y 2
If the domain of f (x) = x 2 is restricted, say to only nonnegative numbers, then the inverse of f is a function.

10. FINDING INVERSES OF NONLINEAR FUNCTIONS
H O R I Z O N T A L L I N E T E S T
If no horizontal line intersects the graph of a function f more than once, then the inverse of f is itself a function.