12.1 Inverse Functions
• For an inverse function to exist, the function
must be one-to-one.
• One-to-one function – each x-value
corresponds to only one y-value and each yvalue corresponds to only one x-value.
• Horizontal Line Test – A function is one-toone if every horizontal line intersects the
graph of the function at most once.
12.1 Inverse Functions
• f-1(x) – the set of all ordered pairs of the
form (y, x) where (x, y) belongs to the
function f.
Note: f -1(x)  1
f ( x)
• Since x maps to y and then y maps back to x
it follows that: f -1 ( f(x))  f 1 ( y)  x
12.1 Inverse Functions
•
Method for finding the equation of the
inverse of a one-to-one function:
y  f (x)
1. Interchange x and y.
2. Solve for y.
3. Replace y with f-1(x)
12.1 Inverse Functions
•
Example: y  ( x  3)3  5
1. Interchange x and y. x  ( y  3)3  5
2. Solve for y. x  5  ( y  3)3
3
x5  y3
3
x 5 3  y
3. Replace y with f-1(x)
f
1
( x)  3 x  5  3
12.1 Inverse Functions
• Graphing inverse functions: The graph of an
inverse function can be obtained by
reflecting (getting the mirror image) of the
original function’s graph over the line y = x
12.2 Exponential Functions
•
Exponential Function: For a > 0 and a not equal
to 1, and all real numbers x, f ( x)  a x
• Graph of f(x) = ax:
1. Graph goes through (0, 1)
2. If a > 1, graph rises from left to right. If 0 < a
< 1, graph falls from left to right.
3. Graph approaches the x-axis.
4. Domain is: (, )
Range is: (0, )
12.2 Exponential Functions
Graph of an Exponential Function
f ( x)  a
(0, 1)
x
12.2 Exponential Functions
•
Property for solving exponential equations:
a a x y
x
•
y
Solving exponential equations:
1. Express each side of the equation as a power
of the same base
2. Simplify the exponents
3. Set the exponents equal
4. Solve the resulting equation
12.2 Exponential Functions
•
Example: Solve: 9x = 27
(3 )  (3)
2 x
3
32 x  33
2x  3  x 
3
2
12.3 Logarithmic Functions
•
Definition of logarithm:
y
y  log a x  x  a
•
Note: logax and ax are inverse functions
•
Since b1 = b and b0 = 1, it follows that:
logb(b) = 1 and logb(1) = 0
12.3 Logarithmic Functions
•
Logarithmic Function: For a > 0 and a not equal
to 1, and all real numbers x, f ( x)  log a x
• Graph of f(x) = logax :
1. Graph goes through (0, 1)
2. If a > 1, graph rises from left to right.
If 0 < a < 1, graph falls from left to right.
3. Graph approaches the y-axis.
4. Domain is: (0, )
Range is: (, )
12.3 Logarithmic Functions
Graph of an Exponential Function
f ( x)  a
x
Try to imagine the inverse function
12.3 Logarithmic Functions
Inverse - Logarithmic Function
f ( x)  log a ( x)
12.3 Logarithmic Functions
•
Example: Solve x = log1255
In exponential form: 5  125 x
In powers of 5:
5  (5 )  5  5
1
3 x
1
3x
Setting the powers equal: 1  3x  x  13
12.4 Properties of Logarithms
•
If x, y, and b are positive real numbers
where b  1
Product Rule: log b xy  log b x  log b y
x
Quotient Rule: log b  log b x  log b y
y
r
log
x
 r log b x
Power Rule:
b
Special Properties: log b b x  x and blogb x  x
12.4 Properties of Logarithms
•
Examples:
Product Rule:
log 3 9 x  log 3 9  log 3 x  2  log 3 x
Quotient Rule: log 1000  log 1000  log y  3  log y
10
10
10
10
y
Power Rule: log 34  4 log 3
5
5
Special Properties: log 12371  371
12
13.1 Additional Graphs of Functions
Absolute Value Function
• Graph of
f ( x)  x
• What is the domain and the range?
13.1 Additional Graphs of Functions
Graph of a Square Root Function
• Graph of f ( x) 
x
(0, 0)
13.1 Additional Graphs of Functions
Graph of a Greatest Integer Function
• Graph of f ( x)  x
Greatest integer
that is less than or
equal to x
13.1 Additional Graphs of Functions
Shifting of Graphs
•
•
Vertical Shifts: y  f ( x)  k
The graph is shifted upward by k units
Horizontal shifts: y  f ( x  h)
The graph is shifted h units to the right
y  a  f (x)
•
•
If a < 0, the graph is inverted (flipped)
If a > 1, the graph is stretched (narrower)
If 0 < a < 1, the graph is flattened (wider)
13.1 Additional Graphs of Functions
• Example: Graph f ( x)  x  4
Greatest integer
function shifted
up by 4
13.1 Additional Graphs of Functions
Composite Functions
• Composite function: function of a function
f(g(x)) = (f  g)(x)
Example: if f(x) = 2x – 1 and g(x) = x2 then
f(g(x)) = f(x2) = 2x2 – 1
• What is g(f(2))?
• Does f(g(x)) = g(f(x))?