12.1 Inverse Relations and
Functions
General Information
 In an inverse relation, the domain of
the original function is the range of
the inverse function
 The range of the original function is
the domain of the inverse of the
function
 The graph of a function and the graph
of its inverse are symmetric with
respect to the line y = x.
Finding the inverse of a RELATION
Given (x, y) : To find the inverse of the relation,
interchange x and y).
The inverse will be (y, x)
Example: Given the relation:
(-3, -4), (2, -5), (-6, 4), (12, 0)
The inverse will be:
(-4, -3), (-5, 2), (4, -6), (0, 12)
Finding the inverse of a
Function/Equation
1. The function must be a one-to-one function.
Every x must have a unique y (range not
repeated)
(Use Horizontal line test)
This is a function,
but NOT a one-to-one
function
2. Notation for an inverse: f-1(x)
3. Process to find the inverse:
a. Change f(x) to y
b. Interchange x & y
c. Solve for y
d. Change y to f-1(x)
Example:
Given f(x) = 3x + 1
Find f-1(x)
f(x) = 3x + 1
y = 3x + 1
x = 3y + 1
x – 1 = 3y
x 1
y
3
x 1
f ( x) 
3
1
-1
Verify that f (x) is an inverse
To verify that f-1(x) is really the inverse, we take the
composite
If f(f-1(x)) = x, then you have the inverse.
f(x) = 3x + 1 and f 1 ( x)  x  1
3
-1
f(f (x)) = f (
=x
x 1
x 1
) = 3
 1
3
 3 
Try this: Find the inverse of the following function
and verify that it is the inverse:
1. f(x) = 2x + 7
Examples
12.2 Exponential Functions
The format of an exponential function is:
f(x) = ax
where “a” is some positive real
number different from 1
a is known as the base
Ex: f(x) = 2x
x
1
g(x) =  
2
The following are NOT exponential
functions:
1
f(x) = x 2
or f(x) = x2
Graph of an exponential function:
Graph of ax
Properties of Exponential functions:
Domain: (, )
Range: (0, )
Increasing
1
Includes 3 points: (0, 1), (1, a), (-1, )
a
Logarithmic Functions
Logarithmic Functions:
A Logarithmic Function is the inverse of
an Exponential Function.
Exponential Function: y  a
Logarithmic Function: x  a
x
y
Logarithmic Function Notation:
log a x
Which reads “log base a, of x”
y  log a x means x=ay
Graphing Logarithmic Functions
Since a logarithmic function is an inverse of an
exponential function, graph the exponential
function then swap the x and y coordinates to
graph the logarithmic function
See p. 524
12.3 Exponential And Logarithmic
Relationships
Converting Exonenential and Logarithmic
Functions:
Recall: y  log a x and x  a y
Convert the following to Logarithmic
equations:
1. 2y = 8
y=log28
2. a-1 = 4
3. 103 = 1000
4. 60 = 1
5. 10-3 = 1000
6.
6
 
5
2

25
36
Convert to Exponential Function:
7. y =log 35
8. -2 = loga 7
9. a = logb d
Solving Logarithmic Equations
Solve the following:
10.log2 x = -3
11.Log27 3 = x
12.log10 x = 4
13.Logx 81 = 4
14.Log2 x = 16
Logarithmic Rules
Theorem 12-3 For any positive number a where
a1
a
and
loga x
x
log a a x  x
Examples:
log4 3
1. 4

2. b
logb 32

7
log
2

3.
2
4. log 2 16 
5. log 3 9 
6. log 4 64 