Algebra Review
Limit at infinity
Notation: If the function values approach a real number L as x increases without
bound, then we write: lim f  x   L .
x
If the function values approach a real number M as x decreases without bound,
then we write: lim f  x   M .
x
If lim f  x   L , then the line y  L is a (rightward) horizontal .
x
If lim f  x   M , then the line y  M is a (leftward) horizontal asymptote.
x
If the graph is given, check the behavior of the given function as x   or as
x   .
Examples:
1 What if the function is defined by a formula?
Example:
1
lim   
x   x 
 1 
lim   
x   x 2 
Fact: For n  0 (a positive rational number)
 1 
lim    0 .
x   x n 
2 In general; a rational function approaches its Horizontal Asymptote as x goes to
infinity (if it has a HA.). To answer limit at infinity questions, think about finding
the HA – compare the degrees of numerator and denominator.
If degree(top)>degree(bottom); no HA.
If degree(top)<degree(bottom); HA is y=0.
If degree(top) = degree(bottom); HA is:
y= leading coefficient of top / leading coefficient of bottom.
Example:
 5x2  x  1 
lim 

x   4 x 2  x 
Example:
 x2  x  1 
lim 

x   x3  2 
Example:
 x  x6 
lim 

x   x3  2 x 
3 A polynomial function does not have a horizontal asymptote; it increases or
decreases without bound as x goes to infinity.
Example:


lim 2 x3  x  1 
x 


lim 5 x 4  x 
x 
For other functions (inverse trig, exponential, logarithmic, etc); you can
simply think about their graphs to answer the limit questions.
Example:
lim  arctan x  
x 
lim
x 
 arctan x  
More on exponentials and log functions in Chapter 3…
4 Basic Functions’ Graphs and Basic Transformations’ Rules
Exponential and Logarithmic Functions
In College Algebra, you should have learned to transform nine basic functions. Here are the
basic functions. You should know the shapes of each graph, domain and range of the function,
and you should be able to state intervals on which the function is increasing and intervals on
which the function is decreasing.
f ( x)  x
f ( x)  x 2
f ( x)  x
f ( x)  3 x
f ( x) 
1
x
f ( x)  e x
f ( x)  x 3
f ( x)  x
f ( x)  ln x
1
You should be able to translate these graphs vertically and/or horizontally, reflect them about the
x or the y axis, and stretch them or shrink them vertically or horizontally.
Vertical Shifting
To graph f ( x)  c,
(c  0) , start with the graph of f ( x) and shift it upward c units.
To graph f ( x)  c,
(c  0) , start with the graph of f ( x) and shift it downward c units.
Horizontal Shifting
To graph f ( x  c),
(c  0) start with the graph of f ( x) and shift it to the left c units.
To graph f ( x  c),
(c  0) , start with the graph of f ( x) and shift it to the right c units.
Reflection of Functions
A reflection is the “mirror-image” of graph about the x-axis or y-axis.
To graph  f ( x) , reflect the graph of f ( x) about the x-axis.
To graph f ( x) , reflect the graph of f ( x) about the y-axis.
Vertical Stretching and Shrinking
Vertical Stretching: If a > 1, the graph of y  af ( x) is the graph of y  f ( x) vertically stretched
by multiplying each of its y -coordinates by a .
Vertical Shrinking: If 0< a < 1, the graph of y  af ( x) is the graph of y  f ( x) vertically shrunk
by multiplying each of its y-coordinates by a.
You may find it helpful to apply transformations in this order:
1.
2.
3.
4.
Vertical and/or Horizontal Stretching or Shrinking
Reflection about the x axis
Horizontal or Vertical translations
Reflection about the y axis
This is not the only order which works, but you will make few mistakes if you apply
transformations in this order.
2
Exponential Functions; The Number e
Functions whose equations contain a variable in the exponent are called exponential functions.
Real-life situations that can be described using exponential functions:
1.
2.
3.
4.
Population growth
Growth of an epidemic
Radioactive decay
Other changes that involve rapid increase or decrease
Definition: The exponential function f with base b is defined by f ( x)  b x (b > 0 and b  1)
and x is any real number.
If b = e (the natural base, e  2.7183), then we have f ( x)  e x , the natural exponential function.
x


 1
Definition: e is the “limiting value” of 1   as x grows to infinity.
 x
It is an irrational number, like  . This means it cannot be written as a fraction nor as a
terminating or repeating decimal.
If b > 1, the graph of f ( x)  b x looks like (larger b results in a steeper graph):
If 0 < b < 1, the graph of f ( x)  b x looks like (smaller b results in a steeper graph):
Both graphs have a horizontal asymptote of y = 0 (the x-axis).
3
Transformations of Exponential Functions
We will apply the same rules we have learned for other functions, but since exponential
functions have a horizontal asymptote we must remember that when the function has a vertical
shift (upward/downward) the horizontal asymptote is shifted by the amount of the vertical shift.
1
Example 1: Sketch the graph of f ( x)   
 3
a. transformations
x 1
 4 . State the:
4
3
2
1
b. domain/range
c. asymptote
-4 -3 -2 -1 -1
-2
-3
-4
d. y-intercept
y
x
1 2 3 4
e. key point
x 1
1
Example 2: Let f ( x)     1
 3
a. State its horizontal asymptote.
 2
b. Is the point  0,
 on the graph of the function?
 3 
c. Is the point (-2, -2) on the graph of the function?
4
Logarithmic Functions
The exponential function is 1-1; therefore, it has an inverse function. The inverse function of the
exponential function with base b is called the logarithmic function with base b.
The function f ( x)  log b x is the logarithmic function with base b with x > 0, b> 0 and b  1.
The Graph of a Logarithmic Function
If b > 1, the graph of f ( x)  log b x looks like:
If 0 < b < 1, the graph of f ( x)  log b x looks like:
Both graphs have a vertical asymptote of x = 0 (the y-axis).
Example 3: For each function, find its domain and vertical asymptote.
a. f ( x)  log 3 ( x  4)
b. f ( x)  ln(3  2 x)
Domain:
Domain:
Vertical Asymptote:
Vertical Asymptote:
5
Transformations of Logarithmic Functions
We will apply the same rules as done for other functions, but since logarithmic functions have a
vertical asymptote we must remember that when the function has a horizontal shift (left or right)
the vertical asymptote is shifted by the amount of the horizontal shift.
Example 4: Sketch the graph of f ( x)   log( x  3) . State the:
a. transformations
4
3
2
1
b. domain/range
c. asymptote
-4 -3 -2 -1 -1
-2
-3
-4
d. y-intercept
y
x
1
2
3
e. key point
Very Important Fact: y  log b x is equivalent to b y  x .
Example 5: Write each equation in its equivalent exponential and/or logarithmic form.
a. 3  log 7 x
b. 2  log b 25
c. log 2 32  5
d. 2 5  x
e. e y  33
Example 6: Evaluate, if possible.
a. log 7 49
b. log 1,000
f. 38  x
c. log 3 27
6
4
d. log 2  8
g. log 2
1
32
j. log 7 7
e. log 0
1
f. log 3  
9
h. ln e10
i. ln e 3
k. log 10
Inverse Property of Logarithms
For b > 0 and b not equal to 1,
1. log b b x  x
2. b log b x  x
Example 7: Evaluate.
a. log18 18 7
b. log 10
c. log   
e
d. 6 log 6 27
e. 51log 51 
f. e ln 100
7
The Product, Quotient and Power Rule
Let b, M and N be positive real numbers with b not equal to 1 and P be any real number.
log b MN  log b M  log b N (Product Rule)
M 
log b    log b M  log b N (Quotient Rule)
N
log b M P  P log b M
(Power Rule)
Example 8: Rewrite each logarithm so that the expression contains no logarithms of products,
quotients or powers.
a. log 100 x 2


 64 
b. log 4  5 
x 
c. log15 (1510  x )
 z 10 x 
d. log 5  5 3 
 3 y 
Example 9: Write the following expression as a single logarithm.
a. log 4 2  log 4 8
b. log 3 6  log 3 2
c. 2 ln x  5 ln x  ln x
8
1
d. log v  log z  4 log w
3
e.  log 2 ( x 2  81)  log 2 ( x  9)
f.  log 5 ( x  5)  log 5 ( x  1)  log 5 ( x  2)
The Change-of-Base Property
For any logarithmic bases a and b, and any positive number M,
log b M 
log a M
.
log a b
The change-of-base property is used to write a logarithm in terms of quantities that can be
evaluated with a calculator. Because calculators contain keys for common (base 10) and natural
(base e) logarithms, we will frequently introduce base 10 or base e.
Example 10: Use natural logarithms to rewrite log17 3 .
Example 11: Use common logarithms to rewrite log 9 11 .
Exponential and Logarithmic Equations
To Solve an Exponential Equation Using Natural Logarithms
1. Isolate the exponential expression.
2. Take the natural logarithm on both sides of the equation.
3. Solve for the variable.
9
Note: The base that is used when taking the logarithm on both sides can be any base at all. If
you use a base other than 10 or e, then you can apply the change-of-base formula to show that
the solutions are the same no matter which base you use.
Example 12: Solve.
a. 4 x  64
b. 3 2 x 1  27
c. 53 x  25 5
d. 2 x 1 
e. 5 x  17
f. 3 x  19
g. 7 x  4
h. 11x  121
1
16
i. 3e 5 x 1  3  1980
10
To Solve Logarithmic Equations
1. Isolate the logarithmic terms on one side of the equation. If necessary, use the properties of
logarithms to write multiple logarithms as a single logarithm.
2. Rewrite the equation in its equivalent exponential form or use the base in the logarithm
equation and raise that base to each side of the logarithmic equation.
Note: You must always check the proposed solutions of a logarithmic equation. Exclude from
this set, solutions that produce the logarithm of a negative number or 0.
Example 13: Solve.
1
a. log16 x 
4
b. ln x  2
c. log 2 (4 x  1)  5  0
d. log 6 ( x  5)  log 6 x  2
e. log 6 (2 x  1)  log 6 ( x  5)
f. log 8 ( x  1)  log 8 ( x  2)  2 log 8 3
g. 13  11 log13 (13 x)  35
11