Inverse Functions

1
Section 5.1

• Inverse Functions

2
Inverses

• For an ordered pair (x, y), the y becomes the x
  and the x becomes the y

• An inverse relation takes the output of the
  original relation and gives back the input of the
  original relation.

3

• Geometrically speaking, an inverse is a
  reflections across the line yx

4
Functions

• Consider then following graphs of functions

Are they functions?
What is a key difference with respect to y?
Why could this matter?
5
One-to-one Functions

• A function is one-to-one if different inputs have
  different outputs

If
OR
When the inputs are the same then the outputs
are the same
If
6
Based on the definitions
Given the function f, prove that f is one-to-one
7
Assume f(a) f(b) for ANY numbers a and b in
the domain of f
Then
So, since ab when f(a) f(b) the function f is
one-to-one
8
One-to-one functions and Inverses
If a function f(x) is one-to-one, then its
inverse f-1(x) is a function
NOTE f-1(x) is a special notation where the -1
is NOT an exponent
The domain and range of a function f(x) switch
with the domain and range of its inverse f-1(x)
A function that is increasing or decreasing over
its domain is one-to-one. Why?
9
What test do we use to see if a graph is that of
a function?
Vertical line test
Is this a function?
Here is that same relations inverse.
Vertical line test
Do you realize a horizontal line reflected across
yx turns into a vertical line?
10
Horizontal line test

• If it is possible for a horizontal line to
  intersect the graph of a function more than once
• then the function is not one-to-one
• and its inverse is NOT a function

11
Finding the formula for inverses
Before you begin, graph the function and do a
horizontal line test. If it fails, go no further
other than stating it fails the HLT and has no
inverse
1. Change f(x) to y
2. Swap x and y
3. Solve for y
4. Replace y with f-1(x)
12
GO
Find the inverse of
13
fails the horizontal line test and does not have
an inverse
is not one-to-one
To prove this we need explain only 1 counter
example
ANY polynomial of even degree will fail the HLT
14
GO
Find the inverse of
Find
If two functions composition yields x they are
inverses of each other That is
then f and g are inverses
15
Special Cases
There are some functions that we need an inverse
for that fail this standard.
In those cases we use a domain restriction on the
inverse to restrict it to a section of the
inverse that will pass the vertical line test.
I bet you thought 2 2 4 too!