Transcript October 3, 2011 At the end of today, you will be able to: Find inverse

September 27, 2012
Inverse of Functions
Warm-up: f(x) = x2 – 1 and g(x)  x
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!
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 1.9
Graphs 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
1
x3
1
f
( x)  x
f (x) 
2
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
x2
f ( x) 
3
1
f(x) = 3x + 2
Show that the
composition of f
and f-1 equals x.
Replace x in f(x)
with (x – 2)/3
 x2
f ( f ( x))  f 

 3 
1
 x2
 3
2
 3 
Simplify
=x–2+2
f(f -1(x)) =
x
YAY!
Now go the other way…
f(x) = 3x + 2
x2
f ( x) 
3
1
f-1(f(x))
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

x 1
g(x) 
7
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 Test
Use 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)