1. 11.4 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. DEF: Inverse Relations f(x) and f-1(x) Switch the x value and the y value
FIND: A-1
A-1= {(-2,-3),(2,-1),(5,3)}
Is A-1 a function?
Yes
P 285
EX A = {(1,2), (2,-3), (5,2)} Find A-1
A-1 = {(2, 1), (-3, 2), (2, 5 )}
EX1A ={(-3,-2),(-1,2),(3,5)} y x 5 -5

3. NOTE: When the inverse of a given function
“f” is a function, f –1 is used to denote it. If a function is defined by an equation, the equation of the inverse is obtained by interchanging x and y in the original equation. P 285
EX: 2 A function “f” is defined by the equation y = -2/3 x Find an equation for f –1(x)
x = -2/3 y + 4
(x = -2/3 y + 4)3
3 x = -2 y + 12
3 x– 12 =-2y
3 x – 12 = y or f-1(x) -2
f-1(x) = - 3x + 6 2 y x 5 -5

4. Absolute Value 5) y = |x| D: 8) x = |y| D: 10) y=|x|+6 D:
x | y x | y x | y
(-1, ) R: (1, ) R: (-1, ) R:
(0, ) (0, ) (0, )
(1, ) F ? (1, ) F ? (1 , ) F? y x 5 -5 y x 5 -5 y x 5 -5

5. Absolute Value 5) y = |x| D: 8) x = |y| D: 10) y=|x|+6 D:
x | y Ʀ x | y [0, ∞) x | y Ʀ
(-1, 1) R: (1, -1) R: (-1, 7) R:
(0, 0) [0, ∞) (0, 0 ) Ʀ (0, 6) [6, ∞)
(1, 1) F ? (1, 1 ) F ? (1 , 7) F?
Yes No Yes y x 5 -5 y x 5 -5 y x 5 -5

6. P286 EX: 3 Graph the function defined by y =  x and its inverse
P286 EX: 3 Graph the function defined by y =  x and its inverse. Is the
inverse a function?
x  x  function inverse
( -2, 2 ) ( 2, -2 )
( -1, 1 ) ( 1, -1 )
( 0, 0 ) ( 0, 0 )
( 1, 1 ) ( 1, 1 )
( 2, 2 ) ( 2, 2 ) y x 5 -5

7. Lines Ax + By = C Standard Form y = mx + b Slope-Intercept Form
9) 5x – 2 y = 10
(0, -5) (2, 0)
2y = -5x + 10
y = 5/2 x – 5
Standard Form
Slope-Intercept Form
Cover-up method y x 5 -5

8. EX:4 Graph the function defined by 4x–2y
=8 and its inverse in the same coordinate
plane. Is its inverse a function?
4x – 2y = 8 Use cover-up (0, ) ( ,0)
(0,-4) (2,0)
4y – 2x = 8 Use cover-up (0, ) ( ,0)
(0,2) (-4,0)
Or get y [f-1(x)] by itself
4y = 2x + 8
f-1(x) or y = ½ x + 2 y x 5 -5

9. NOTE: If a function is defined by an equation in the form y = mx + b, m  0, then its inverse can be written in the same form, and the inverse is a function. y x 5 -5
P297 f ( x ) = 3 x – 5
or y = 3 x – 5
x = 3y – 5
x + 5 = 3y
x + 5 = y [f-1(x)] 3
29) f –1 ( 1 )
(1 + 5)/3 = 2
30) f –1 ( 4 )
(4 + 5)/3 = 3
31) f –1 ( -5 )
(-5 + 5)/3= 0
32) f –1 ( 16 )
(16 +5)/3= 7
OR 1 = 3x – 5; 6 = 3x; 2 = x
OR 4 = 3x – 5; 9 = 3x; 3 = x
OR -5 = 3x – 5; 0 = 3x; 0 =x
OR 16 = 3x – 5; 21 = 3x; 7 = x

10. 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

11. 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

12. 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).

13. 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.

14. 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.

15. ( ) 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

16. 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.

17. 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.

18. 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.

19. Modeling with an Inverse Function
ASTRONOMY Near the end of a star’s life the star will eject gas, forming a planetary nebula. The Ring Nebula is an example of a planetary nebula.
The volume V (in cubic kilometers) of this nebula can be modeled by V = (9.01 X ) t 3 where t is the age (in years) of the nebula. Write the inverse function that gives the age of the nebula as a function of its volume.

20. Modeling with an Inverse Function
Volume V can be modeled by V = (9.01 X ) t 3
Write the inverse function that gives the age of the nebula as a function of its volume.
SOLUTION
V = (9.01 X 1026 ) t 3
Write original function. V
9.01 X 10 26
= t 3
Isolate power. V
9.01 X 10 26 3
= t
Take cube root
of each side.
Simplify.
(1.04 X 10– 9 ) V = t 3

21. The Ring Nebula is about 5500 years old.
Modeling with an Inverse Function
Determine the approximate age of the Ring Nebula given that its volume is about 1.5 X cubic kilometers.
SOLUTION
To find the age of the nebula, substitute 1.5 X for V.
t = (1.04 X 10– 9 ) V 3
Write inverse function.
= (1.04 X 10– 9 ) X 1038 3
Substitute for V.
 5500
Use calculator.
The Ring Nebula is about 5500 years old.