7.8 Inverse Functions and
Relations
Horizontal line Test
Look at the functions f(x) and g(x)
f(x) = 2x + 4
g(x) = ½x – 2
(x, f(x))
(x,g(x))
(-1,2)
(2,-1)
(0,4)
(4,0)
(1,6)
(6,1)
(2,8)
(8,2)
What do you see?
Look at the functions f(x) and g(x)
f(x) = 2x + 4
g(x) = ½x – 2
(x, f(x))
(x, g(x))
(-1,2)
(2,-1)
(0,4)
(4,0)
(1,6)
(6,1)
(2,8)
(8,2)
The two functions are inverses of each other
Definition of Inverse Functions
A function and its inverse function can be
described as the "DO" and the "UNDO"
functions. A function takes a starting
value, performs some operation on this
value, and creates an output answer. The
inverse function takes the output answer,
performs some operation on it, and arrives
back at the original function's starting
value.
http://www.regentsprep.org/Regents/math/algtrig/ATP8/inverselesson.htm
How to find an inverse function
Since the input and output switch places.
x and y will switch places.
Function
Inverse
y = 4x +12
x = 4y + 12
4y = x- 12
y = ¼x – 3
Find the inverse of y = 5x - 20
Switch x and y
x  5 y  20
Find the inverse of y = 5x - 20
Switch x and y
x  5 y  20
x  20  5 y
Find the inverse of y = 5x - 20
Switch x and y
x  5 y  20
x  20  5 y
1
x4 y
5
1
y x4
5
Graph the function and it inverse
y  5 x  20
1
y x4
5
The graphs the function
and its inverse reflect over a line
y=x
y  5 x  20
1
y x4
5
To check if two functions are
inverse we use compositions
Let
3
f x   x  6
4
4
g ( x)  x  8
3
If both compositions equal x, then the
functions are inverses f  g  x  
g  f  x  
To check if two functions are
inverse we use compositions
If both compositions equal x, then the
functions are inverses f ( g ( x))  3  4 x  8   6
3
f x   x  6
4
4
g ( x)  x  8
3
43

 x66
x
43

g ( f ( x))   x  6   8
34

 x 88
x
Inverses can be written as y and y
If both compositions equal x, then the
functions are inverses f ( g ( x))  3  4 x  8   6
3
y  x6
4
4
y  x  8
3
43

 x66
x
43

g ( f ( x))   x  6   8
34

 x 88
x
Horizontal Line test
If a Horizontal line can pass through a graph
of a function only touching it at one point,
then the graph has a inverse.
Yes
No
Homework
Page 393 –
# 15, 21, 24,
27, 30, 33,
36
Homework
Page 393 –
# 18, 20, 23,
26, 29, 32,
35