September 27, 2012Inverse of Functions
Warm-up: f(x) = x2 – 1 and
Find the following compositions, then state the domain
1. (f o g)(x) 2. (g o f)(x)
CW 1.9: Pg. 99 #15-23odd, Pg. 90 #35, 37
Test Monday/Tuesday!
g(x) x
What are inverses? f-1(x)• Inverse of multiplication is__________________
• Inverse of addition is______________________
• Inverse of a square root is__________________
• Inverse of squared is______________________
• Inverse of the relation {(-5, 4), (-1, 5), (0, 2), (3, 4)} is:
Lesson 1.9Graphs of Inverses – What do you notice?Make a table for each and graph their points. What do you notice about their points? f(x) = 2x – 3 f(x) = x2, x ≥ 0
f 1(x) x 32
xxf )(1
Finding the Inverse Function, f -1(x), algebraically
Find the inverse function of: f(x) = 3x + 2
1) Rewrite f(x) to y
2) Switch the x and y variables.3) Solve for y
Show that the two functions are inverses algebraically and graphically
f (x) x 4
g(x) x 2 4, x 0
The composition of a function and its inverse will always equal x.
Let f and g be two functions:Two functions are inverses if and only if:
(f o g)(x) = x and (g o f)(x) = x
Verifying that the two functions are inverses: by using the composition, f(f -1(x)) = x
f(x) = 3x + 23
2)(1
x
xf
23
23
x
3
2))(( 1 x
fxff
= x – 2 + 2
f(f -1(x)) = x
Show that the composition of f and f-1 equals x.
Replace x in f(x) with (x – 2)/3
SimplifyYAY!
Now go the other way…
f(x) = 3x + 2
f-1(f(x))
3
2)(1
x
xf
Ways to verify two functions are inverses
• The compositions of the two functions equal x: (f o g)(x) = x and (g o f)(x) = x
• The graph of the inverse function is a reflection of the graph of f over the y = x line.
• If the coordinates of the function are (a, b), the inverse function coordinates are (b, a).
Are the following functions two inverses of each other? Show/explain how to check. f(x) = 1 + 7x and
g(x) x 17
The inverses of function A and D are functions, but B and C are not. Why? Figure out a rule or a test
that tells you whether or not it is a function.
A B
C D
Fill out the chart to help organize our Unit 1 TestUse f(x) = x2 – 9 to find the following
Zeros Domain Use a graph to find the range and determine where it is increasing, decreasing, and/or constant..
Inverse Composition of function and inverse
Evaluating f(2x – 3)