Math 3 Honors: Unit 5
Name: ______________________________
INVERSE FUNCTIONS
Inverses f-1(x): Relation (function) where you
switch the domain and range values
Points of f(x) means (x, f(x)) or (x, y)
Points of f-1(x) means (f(x), x) or (y, x)
Domain of Function f(x)
Range of Inverse f-1(x)
Range of Function f(x)
Domain of Inverse f-1(x)
Properties of Inverses:
Function, f(x): A function is a relation so that
every x in the domain has just one y.
If f(x) and g(x) are inverses of each other, g(x) = f-1(x) and f(x) = g-1(x),
Property #1: f(a) = b if and only if g(b) = a
f(a) = b means (a, b) and g(b) = a means (b, a)
Determine if f(x) and g(x) are inverses:
1. f(x) = {(3, 2), (-4, 1), (-1, 5), (9, 3) }
2. f(x) = {(-3, 5), (-6, 2), (-7, 8), (-4, -4)}
g(x) = {(1, -4), (5, -1), (2, 3), (3, 9)}
g(x) = {(8, -7), (5, -3), (2, 6), (-4, -4)}
Property #2: f ( g( x ))  x and g( f ( x ))  x
Composition of two functions will always result in an output of x.
Determine if f(x) and g(x) are inverses:
3. f ( x )  2 x  4
g( x ) 
5.
1
x2
2
f ( x)  x 2  2
g( x ) 
4. f ( x )  4 x  24
x2
g( x ) 
6.
1
x6
4
f ( x) 
x 1
g( x )   x  1
2
How do you find the inverse of a function?
By the graphs: Switch all points (x , y) and graph the new points.
Find the inverse of the given function:
7. f(x) = {(1, 3), (2, -5), (3, 6)}
8. g(x) = {(-4, 1). (-3, 2), (0,0), (1, 10)}
By the equation: Change x and y; solve for y to find inverse.
Given y = f(x)  Change x and y: x = f(y)  Solve for y to get inverse f-1(x)
Find the inverse of the given function:
9. f ( x )  3 x  7
12. f ( x ) 
15.
2x  8
f ( x)  3 x  4
10. g( x ) 
13. g ( x ) 
2
x6
5
3x  6
4 3
16. g( x )  x  2
5
11. h( x ) 
3x  4
2
14. h( x ) 
x2
x
17. h( x )  ( x  5)  2
2