Inverse Functions

1
Inverse Functions Logarithms

• See section 1.6 in the text

2
Which functions have inverse functions?

• Example. Squaring function with various domains
• All real numbers
• Nonnegative real numbers
• Non-positive real numbers
• One-to-one Function f( x1 ) ? f( x2 ) whenever
  x1 ? x2

3
Relationships between function its inverse

• Dual equations
• y f(x) ? f -1(y) x
• Graphs mirror at the line y x (example)
• Domains and ranges
• Cancellation properties

4
Example mirror property
5
Example mirror property
6
Finding formula for inverse of a one-to-one
function

• Write down the equation y f(x)
• Solve this equation for x in terms of y, if
  possible
• Formula in (2) is inverse. Express as function
  of x by interchanging x and y.
• Example.

7
Logarithms as Inverse Functions

• Dual equations
• y ax ? loga(y) x
• Graphs mirror at the line y x
• Domains and ranges
• Cancellation properties
• loga(ax ) x for real nos. x
• a loga(x) x for x gt 0

8
More on Logarithms

• Students should review the Laws of Logarithms
  (see p. 68). Know them.
• Special logarithms - common and natural bases

9
Inverse Trigonometric Functions

• Restrict domains to get one-to-one functions.
• Look at sine and cosine.
• Composition properties

10
The Arcsine Function

• Restrict domain of sine function to get
  1-1function

11
The Arcsine Function

• Reflection property of graphs

12
A Closer Look at the Arcsine

• Domain
• Range
• Composition examples

13
More on Compositions
14
A Closer Look at the Arccosine

• Domain
• Range