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
DEF: Inverse Relations f(x) and f-1(x)
Switch the x value and the y value
P 285
EX A = {(1,2), (2,-3), (5,2)}
A-1
Find
A-1 = {(2, 1), (-3, 2), (2, 5 )}
EX1A ={(-3,-2),(-1,2),(3,5)}
NOTE: When both a
relation and its inverse
are functions, they are
Called inverse functions.
FIND: A-1
A-1= {(-2,-3),(2,-1),(5,3)}
Is A-1 a function?
Yes
5
y
x
-5
-5
5
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 + 4. 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
5
y
x
-5
-5
5
P286 EX: 3 Graph the function defined by
y =  x and its inverse. Is the
inverse a function?
5
x
-2
x 
function
inverse
_______________________________
2
( -2, 2 )
( 2, -2 )
______________________________
1
( -1, 1 )
( 1, -1 )
_________________________
0
( 0, 0 )
( 0, 0 )
______________________________
1
( 1, 1 )
( 1, 1 )
______________________________
2
( 2, 2 )
( 2, 2 )
-5
-5
-1
0
1
2
y
x
5
Absolute Value
5) y = |x| D: 8) x = |y|
x|y
x|y
(-1, )
(0, )
(1, )
5
R:
F?
y
(1, )
(0, )
(1, )
5
D: 10) y=|x|+6 D:
x|y
R:
F?
y
5
x
-5
-5
5
(-1, )
(0, )
(1 , )
R:
F?
y
x
-5
-5
5
x
-5
-5
5
Absolute Value
5) y = |x| D: 8) x = |y|
x|y Ʀ
x|y
(-1, 1)
(0, 0)
(1, 1)
5
R:
(1, -1)
[0, ∞) (0, 0 )
F ? (1, 1 )
Yes
y
5
y
D: 10) y=|x|+6 D:
[0, ∞) x | y Ʀ
R:
Ʀ
F?
No
x
-5
-5
5
(-1, 7) R:
(0, 6) [6, ∞)
(1 , 7)
F?
Yes
5
y
x
-5
-5
5
x
-5
-5
5
Lines
Ax + By = C
y = mx + b
9) 5x – 2 y = 10
(0, -5) (2, 0)
2y = -5x + 10
y = 5/2 x – 5
5
y
x
Standard Form
Slope-Intercept Form
Standard Form
Cover-up method
Slope-Intercept Form
P286
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
5
-5
-5
y
x
5