Inverse Functions

1
Section 4.1

• Inverse Functions

2
Inverse Relations

• Interchange coordinates of each ordered pair in
  the relation.
• If a relation is defined by an equation,
  interchange the variables.

3
Important!

• Graphs of a relation and its inverse are always
  reflections of each other across the line y x.

4
Functions and One-to-One Functions

• Recall that a function exists when domain values
  do not repeat. (x-values dont repeat).
• A function is one-to-one if the y-values do not
  repeat.

5
One-to-One Function
A function f is a one-to-one function if, for two
elements a and b from the domain of f, a b
implies f(a) f(b)
x-values do not share the same y-value
6
Horizontal Line Test

• Use to determine whether a function is one-to-one
• A function is one-to-one if and only if no
  horizontal line intersects its graph more than
  once.

7
Why are one-to-one functions important?

• One-to-One Functions
• have
• Inverse Functions!
• If the function is not one-to-one, then the
  inverse is NOT a function.

8
Inverses of Functions

• If the inverse of a function f is also a
  function, it is named f ?1 and read f-inverse.
• The negative 1 in f ?1 is not an exponent. This
  does not mean the reciprocal of f.
• f ?1(x) is not equal to

9
How do we tell if a function is one-to-one?

• A function f is one-to-one if different
  inputs have different outputs.
• That is,
• if a ? b
• then f(a) ? f(b).

• A function f is one-to-one if when the
  outputs are the same, the inputs are the same.
• That is,
• if f(a) f(b)
• then a b.

10
Properties of One-to-One Functions and Inverses

• If a function is one-to-one, then its inverse is
  a function.
• The domain of a one-to-one function f is the
  range of the inverse f ?1.
• The range of a one-to-one function f is the
  domain of the inverse f ?1.
• A function that is increasing over its domain or
  is decreasing over its domain is a one-to-one
  function.

11
How to find the Inverse of a One-to-One Function

• Replace f(x) with y in the equation.
• Interchange x and y in the equation.
• Solve this equation for y.
• Replace y with f-1(x).
• Any restrictions on x or y should be considered.
• Remember Domain and Range are interchanged
• for inverses.

12
Example

• Determine whether the function
• f(x) 3x ? 2 is one-to-one, and if it is,
• find a formula for f ?1(x).

13
Graph of Inverse f-1 function

• The graph of f-1 is obtained by reflecting the
  graph of f across the
• line y x.
• To graph the inverse f-1 function
• Interchange the points on the graph of f to
  obtain the points on the graph of f-1.
• If (a,b) lies on f, then (b,a) lies on f-1.

14
Example

• Graph f(x) 3x ? 2 and
• f ?1(x)
• using the same set of axes.
• Then compare the two graphs.

15
Solution
f ?1(x)
16
Inverse Functions and Composition

• If a function f is one-to-one, then f ?1 is
  the unique function such that each of the
  following holds
• for
  each x in the domain of f,
  and
• 
  
• for each x in the domain of f ?1.

17
Example of how to show work!!

• Given that f(x) 7x ? 2, use composition of
  functions to show that f ?1(x) (x 2)/7.
• Solution

18
Restricting a Domain

• When the inverse of a function is not a function,
  the domain of the function can be restricted to
  allow the inverse to be a function.
• In such cases, it is convenient to consider
  part of the function by restricting the domain
  of f(x). If the domain is restricted, then its
  inverse is a function.

19
Restricting the Domain
Recall that if a function is not one-to-one,
then its inverse will not be a function.
20
Restricting the Domain
If we restrict the domain values of f(x) to those
greater than or equal to zero, we see that f(x)
is now one-to-one and its inverse is now a
function.