7.8 inverse functions and relations horizontal line test
TRANSCRIPT
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 205 yx
Find the inverse of y = 5x - 20
Switch x and y
yx
yx
520
205
Find the inverse of y = 5x - 20
Switch x and y
45
1
45
1
520
205
xy
yx
yx
yx
Graph the function and it inverse 205 xy 4
5
1 xy
The graphs the function and its inverse reflect over a line
y=x
205 xy 45
1 xy
To check if two functions are inverse we use compositions
Let
If both compositions equal x, then the functions are inverses
64
3 xxf 8
3
4)( xxg
xfg
xgf
To check if two functions are inverse we use compositions
If both compositions equal x, then the functions are inverses
64
3 xxf
83
4)( xxg
x
x
xxfg
x
x
xxgf
88
864
3
3
4))((
66
683
4
4
3))((
Inverses can be written as
If both compositions equal x, then the functions are inverses
64
3 xy
83
4 xy
x
x
xxfg
x
x
xxgf
88
864
3
3
4))((
66
683
4
4
3))((
yandy
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