september 27, 2012 inverse of functions warm-up: f(x) = x 2 – 1 and find the following...

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)