HARTFIELD – PRECALCULUS
Unit 1 Functions, Exponentials, and Logarithms
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Function Basics
Even and Odd Functions
Basic Functions
Transformations
One-to-One Functions
Inverse Function
Finding an Inverse Function
Graphs of Inverse Functions
Exponential Functions
The number e
Logarithms
Special Logarithms
Change of Base Formula
Logarithmic Functions
Laws of Logarithms
Solving Exponential & Logarithmic Equations
UNIT 1 NOTES | PAGE 1
Know the meanings and uses of these terms:
Even or Odd Function
Transformation (Translation, Reflection, Dilation)
One-to-one function
Inverse function
Exponential expression, exponential function
Logarithmic expression, logarithmic function
Base of an exponential or logarithmic expression
Exponent of an exponential or logarithmic expression
Common logarithm
Natural Logarithm
Extraneous solution
Review the meanings and uses of these terms:
Function
Domain
Range
Asymptote
HARTFIELD – PRECALCULUS
Function Basics
Definitions: A function f is a rule that assigns each
element x in a set A to exactly one
element, called f(x), in a set B.
Set A is called the domain.
The domain of a function f is the
collection of values which are
acceptable inputs for f.
Set B is called the range.
The range of a function f is the
collection of values which are
possible outputs for f.
The variable of the range, as notated
f(x), can be read as the value of f at x
or as the image of x under f.
UNIT 1 NOTES | PAGE 2
An example of a function written in symbolic
notation could be:
g(t )  t  1, t  0
2
The variable of the domain is t. It is written inside
parentheses on one side of the function.
The name of the function is g. The name is always
written in front of the variable of the domain.
The variable of the range is g(t). It is formed by
the name and variable of the domain.
The rule of the function is t² + 1. The rule is an
expression that finds g(t) from t.
The condition of the function is t > 0. If a function
has a condition, it always follows the rule and it
defines the domain of the function.
HARTFIELD – PRECALCULUS
Identify the parts of a function from the example
below. Then evaluate the function at x = 2.
Ex.:
UNIT 1 NOTES | PAGE 3
When a function is evaluated, it corresponds to a
pair of coordinates on the coordinate plane. In
this example, the ordered pair is (x, H(x)).
H(x) = (x – 4)², x < 4
The graph of the function can be created by all the
possible ordered pairs formed by evaluating the
function.
HARTFIELD – PRECALCULUS
Even and Odd Functions
A function is even if evaluating the function at its
opposite is equal to the original function.
Symbolically we would say f is even if f(−x) = f(x).
UNIT 1 NOTES | PAGE 4
Determine whether each function is even, odd, or
neither even nor odd.
Ex. 1: f (x)  6 x  2 x
A function is odd if evaluating the function at its
opposite is equal to the opposite of the original
function. Symbolically we would say f is odd if
f(−x) = −f(x).
Even functions display symmetry with respect to
the y-axis. Odd functions display symmetry with
respect to the origin.
Many functions are neither even nor odd.
9x
Ex. 2 g(x)  3
x  5x
3
HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 5
Basic Functions
f (x)  x
A function formed by a simple rule applied to a
variable can be a basic function.
The absolute value function
Important basic functions include:
The first-degree reciprocal function f (x)  1 x
The squaring function
Domain:  ,  
Range:
f (x)  x
0, 
The cubing function
Domain:  ,  
Range:
f (x)  x 3
 ,  
The square root function
Domain:  0, 
Range:
f ( x)  x
0, 
The cube root function
Domain:  ,  
Range:
f ( x)  3 x
 ,  
2
Domain:
Domain:
Range:
 ,  
Range:
0, 
 ,0   0,  
 ,0   0,  
Squaring and cubing functions are examples of
power functions. All power functions either
behave similarly to a squaring function (if evenpowered) or a cubing function (if odd-powered).
All root functions either behave similarly to
square root functions (if even-indexed) or cube
root functions (if odd-indexed).
A handout at the professor’s web page is provided
to show the basic functions in greater detail.
HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 6
Transformations
Vertical Translations
Definition: A transformation is an operation
applied to a function that changes
the size, direction, or position of the
graph of the function.
Vertical Translations occur when a number is
added outside of an existing function:
Three transformations can be performed by
applying algebra:
Translations
“shift the graph”
Reflections
“flip the graph”
Dilations
“stretch the graph” or
“compress the graph”
change the position of a graph
requires addition
y  f ( x)  k
If k is positive,
the graph of f is shifted up by k units.
If k is negative,
the graph of f is shifted down by k units.
change the direction of a graph
requires negation
change the size of a graph
requires multiplication
y  x2
y  x2  2
y  x2  2
Note: If a function has multiple transformations,
vertical translations should occur last.
HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 7
Horizontal Translations
Reflections
Horizontal Translations occur when a number is
subtracted inside of an existing function:
Reflections occur when either the independent
variable or the basic function is negated:
y  f (x  h)
y   f (x) causes
the graph of f to be flipped over the x-axis.
If h is positive (which preserves the – inside),
the graph of f is shifted right by h units.
y  f ( x) causes
the graph of f to be flipped over the y-axis.
If h is negative (which produces a + inside),
the graph of f is shifted left by h units.
y x
y  x 3
y  x 3
Note: If a function has multiple transformations,
horizontal translations should occur first.
y x
y  x
y  x
HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 8
Vertical Dilations
Horizontal Dilations
Vertical Dilations occur when a number is
multiplied outside of an existing function:
Horizontal Dilations occur when a number is
multiplied inside of an existing function:
y  c  f (x)
y  f (c  x)
If c  1,
the graph of f is stretched away from the x-axis
by a factor of c .
If c  1,
the graph of f is stretched away from the y-axis
by a factor of 1c .
If c  1,
the graph of f is compressed toward the x-axis
by a factor of c .
If c  1,
the graph of f is compressed toward the y-axis
by a factor of 1c .
y3x
y  33 x
Vertically stretched
y
13
2
x
Vertically compressed
y  x3
y   x
y   2x 
Horizontally stretched
Horizontally compressed
1
3
3
3
HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 9
Given the graph of the function below, use the
coordinate plane to determine the graphs of the
transformed functions.
Given the graph of the function below, use the
coordinate plane to determine the graphs of the
transformed functions.
Ex. 1a:
Find f(x) – 3
Ex. 2a:
Find g(x – 3)
Ex. 1b:
Find –f(x + 1)
Ex. 2b:
Find 2 g(x) + 1
g
f
HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 10
Use basic functions and transformations to
determine the domain and range of each given
function.
Use basic functions and transformations to
determine the domain and range of each given
function.
Ex. 1: g(x)   x  3  5
Ex. 3: F (x)   x  2  3
4
6
Ex. 2: f (x) 
x 2
Ex. 4: G(x)  4  5  x
2
HARTFIELD – PRECALCULUS
One-to-One Functions
Definition: A function f is said to be one-to-one if
for every value f(x) in the range of f
there is exactly one corresponding xvalue in the domain of f.
UNIT 1 NOTES | PAGE 11
Use the horizontal line test to determine if the
following functions are one-to-one.
4
Ex. 1: f (x) 
6
x 2
Ex. 2: g(x)   x  3  5
2
The horizontal line test can be used to determine
whether a function is one-to-one. A function is
one-to-one only if every horizontal line applied to
the graph of the function intersections at most
once.
Ex. 3: h(x)   x 3  7
HARTFIELD – PRECALCULUS
A function that is one-to-one over its natural
domain may be one-to-one over a limited domain.
For example, the function F (x)   x  2  3 is not
one-to-one over its entire domain. However
F1 (x)   x  2  3, x  2 and
F2 (x)   x  2  3, x  2 are both one-to-one
(based on the condition added to F that limits the
domain) without changing the range.
UNIT 1 NOTES | PAGE 12
Given a function that is not one-to-one over its
natural domain, determine a condition that could
be added so that the function becomes one-toone without changing the range.
Ex:
f  x     x  1  2
2
HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 13
Inverse Functions
Property of Inverse Functions
Definition: An inverse function is a one-to-one
function such that if a one-to-one
function f with domain A and range B
−1
exists, then the inverse function f
(read as “f inverse”) has domain B
and range A and the following
property is satisfied:
For two functions f and g to be inverse functions
of each other, the following conditions must be
established:
f x  y  f
1
y   x
1. f and g must be one-to-one functions over
their given domains,
2. the domain of f must be the same as the
range of g and the domain of g must be the
same as the range of f, and
for all x in A and all y in B.
3. the composition of f and g must be x
HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 14
Determine if functions f and g are inverse
functions of each other.
Ex.:
f x  x 3
Finding an Inverse Function
g  x    x  3 , x  3
To find the inverse function of a given function f:
2
1. Determine if f is one-to-one.
2. Find the domain and range of f.
3. Write an equation where the variable of
function is represented by y.
4. Exchange the independent variable for y and
vice versa, creating an inverse statement.
5. Solve for y.
6. Resolve any issues with the rule to ensure that
f −1 is one-to-one and has the appropriate
domain and range.
HARTFIELD – PRECALCULUS
Find the inverse function of f.
Ex. 1: f  x    x 2  1, x  0
UNIT 1 NOTES | PAGE 15
Find the inverse function of f.
Ex. 2: f  x    x  4  2
HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 16
Graphs of Inverse Functions
Since inverse functions effectively reverse the
roles of the independent and dependent
variables, the graph of an inverse function should
satisfy the following pair of properties:
1. For every point (a, b) on the graph of f, the
−1
graph of f should include a point (b, a).
−1
2. The graph of f should demonstrate
symmetry to f with respect to the line y = x.
Find the graph of the inverse of the function
graphed below.
Ex.:
HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 17
Exponential Functions
Definition: The exponential function with base a,
such that a is a positive real number
other than 1, is defined by
f  x   a x , a  0.
Domain:
 ,  
Key Point: (0, 1)
Range:
Examples:
f  x   2x
 0,  
Asymptote: y = 0
f x  

4 x
5
Recall that an asymptote is a line that a graph
approaches as values grow without bound.
If the base a > 1, the function will increase over its
domain with asymptotic behavior as x → −∞.
If the base a < 1, the function will decrease over
its domain with asymptotic behavior as x → ∞.
A special number frequently associated with
exponential functions is e.
HARTFIELD – PRECALCULUS
The number e
Definition: The natural exponential base is the
number e, such that
n
 1
e  lim  1    2.71828182846 .
n 
n
UNIT 1 NOTES | PAGE 18
Identify the base of the exponential function and
use transformations to determine the domain,
range, and asymptote of each given function.
Ex. 1: f (x)  4 x 1  2
An exponential function with a base
of e is called a natural exponential
x
function: f  x   e .
Ex. 2: g(x)  2e x 3
HARTFIELD – PRECALCULUS
Logarithms
The inverse of an exponential expression is a
logarithm.
Definition: Let a be a positive real number other
than 1. Then the logarithm with base
a, denoted as loga, is defined as
follows: loga x  y  a y  x.
UNIT 1 NOTES | PAGE 19
All logarithmic expressions satisfy one of the
following properties:
loga 1  0,
loga a  1,
Evaluate.
Ex. 1: log6 1
Ex. 2: log8 8
More specific to above, the inverse of an
exponential expression with base a is a logarithm
with base a.
Thus the value of a logarithmic expression is equal
to the exponent of base a which equals the input.
Ex. 3: log7 49
loga an  n
HARTFIELD – PRECALCULUS
Rewrite each statement into its inverse form.
Ex. 1:
34 = 81
2
Ex. 2:
7 = 49
Ex. 3:
5x = 20
UNIT 1 NOTES | PAGE 20
Special Logarithms
Two special logarithms can be written without a
base.
Definition: The common logarithm is the
logarithm with base 10 such that
log10 x = log x.
Ex. 4:
log6 36 = 2
Definition: The natural logarithm is the
logarithm with base e such that
loge x = ln x.
Ex. 5:
log2 64 = 6
(Note:
Ex. 6:
log11 x = 2
In some higher levels of mathematics,
log x may actually refer to a natural
logarithm.)
HARTFIELD – PRECALCULUS
Evaluate. Approximate as necessary to five digits.
Ex. 1: log 1000
Ex. 2: log 0.01
UNIT 1 NOTES | PAGE 21
Rewrite each statement into its inverse form.
Ex. 1:
10x = 500
Ex. 2:
ex = 20
Ex. 3:
log x = 3
Ex. 4:
ln x = 5
Ex. 3: ln e
A calculator may be helpful in approximating
some logarithmic expressions:
Ex. 4: log 400
Ex. 5: ln 40
HARTFIELD – PRECALCULUS
Change of Base Formula
Many logarithms may not have an obvious
exponential value. Common and natural
logarithms can easily be used to approximate
exponential values but a second layer approach is
necessary with other bases.
log b m ln m log m
loga m 


log b a lna log a
UNIT 1 NOTES | PAGE 22
Evaluate. Approximate as necessary to five digits.
Ex. 1:
log 4 100
Ex. 2:
log2 0.5
Ex. 3:
log 2 3 12
HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 23
Logarithmic Functions
Definition: The logarithmic function with base a,
such that a is a positive real number
other than 1, is defined by
f  x   logax, a  0.
Domain:
 0,  
Key Point: (1, 0)
Range:
 ,  
Asymptote: x = 0
If the base a > 1, the function will increase over its
domain with asymptotic behavior as x → 0.
If the base a < 1, the function will decrease over
its domain with asymptotic behavior as x → 0.
Observe that the definition of a logarithmic
function, in conjunction with the third property of
logarithms, satisfies the Property of Inverse
Functions.
That is, a logarithmic function of base a is the
inverse function to an exponential function of
base a.
Whereas the asymptote of an exponential
function is horizontal, the asymptote of a
logarithmic function is vertical. Further, the
logarithmic function has an x-intercept instead of
a y-intercept like the logarithmic function.
To sketch the graph of a logarithmic function not
base 10 or e, use the change of base formula.
HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 24
ln x
Example: f  x   2 
ln2
x
Furthermore, since logarithmic functions and
exponential functions are
inverses, they satisfy the
graphical properties of
inverse functions as
illustrated at right with
the natural exponential
function and the natural
logarithmic function.
Identify the base of the exponential function and
use transformations to determine the domain,
range, and asymptote of each given function.
Ex. 1: f (x)  2log 5 (x  1)
Ex. 2: g(x)  2  log  3  x 
HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 25
Laws of Logarithms
The laws of logarithms allow us to rewrite
logarithmic expressions so they are easier to
manipulate. Each law of logarithms has an
analogue with the laws of exponents.
Name
Law of Logarithms
Rewrite each logarithm into a sum or difference of
logarithms so that no logarithm consists of a
product, quotient, or power (where possible).


Ex. 1:
log3 x 4 y 2
Ex. 2:
log 5 x 2  1
Law of Exponents
Product-to-Sum
loga  xy   loga x  loga y
a m  a n  a m n
Quotient-to-Difference
x
loga    loga x  loga y
y
am
m n

a
n
a
Power-to-Product
 
loga x
n
 nloga x
 
a
m n
 am  n


HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 26
Rewrite each logarithm into a sum or difference of
logarithms so that no logarithm consists of a
product, quotient, or power (where possible).
 3x 
log  2 
 yz 
5
Ex. 3:
Ex. 4:
log 4
x 2
4 x3
Rewrite each as a single logarithm. Simplify
where possible.
Ex. 1:
2log6 x  4log6 y  log6 z
Ex. 2:
log2 x  2log2 y  5log2 z
Ex. 3:
2ln3  4ln2
Ex. 4:
3ln5  2ln6
HARTFIELD – PRECALCULUS
UNIT 1 NOTES | PAGE 27
Solving Exponential Equations
Solving Logarithmic Equations
Procedure for most exponential equations:
Procedure for logarithmic equations:
1. Isolate an exponential expression on one side.
2. Take the natural logarithm (or common
logarithm) of both sides.
3. Use the laws of logarithms to rewrite the
exponential expression so that no variable
remains in the exponent.
4. Apply basic algebraic and arithmetic
manipulation to solve for x.
5. Use the laws of logarithms to rewrite the
solution as appropriate and approximate the
solution.
6. Check your solution.
1. Use the laws of logarithms to combine
logarithms on each side as necessary.
2. Apply next step based on the structure:
a. If only one side of the equation has a
logarithm, rewrite the equation into
exponential form.
b. If both sides of the equation have
logarithms and the base is the same, set the
arguments equal to each other.
3. Apply basic algebraic and arithmetic
manipulation to solve for x.
4. Check your solutions. Like radical equations,
logarithmic equations can (and often do)
produce extraneous solutions.
Some exponential equations may use other
procedures for part or all of the entire process; for
example: quadratic-type equations, et.al.
HARTFIELD – PRECALCULUS
Solve the equation. Write the solution as a single
logarithmic expression and approximate the
solution to five places.
Ex. 1:
5x2  20
UNIT 1 NOTES | PAGE 28
Solve the equation. Write the solution as a single
logarithmic expression and approximate the
solution to five places.
Ex. 2:
4  102 x  60
HARTFIELD – PRECALCULUS
Solve the equation. Write the solution as a single
logarithmic expression and approximate the
solution to five places.
Ex. 3:
UNIT 1 NOTES | PAGE 29
Solve each equation.
Ex. 4:
4 x 1  42 x 1
Ex. 5:
43 x 3  8 x 3
3e1 x  60
HARTFIELD – PRECALCULUS
Solve the equation. Write the solution as a single
logarithmic expression and approximate the
solution to five places.
Ex. 6:
53 x  22 x 1
UNIT 1 NOTES | PAGE 30
HARTFIELD – PRECALCULUS
Solve the equation. Write the solution as a single
logarithmic expression and approximate the
solution to five places.
Ex. 7:
32 x 1  62 x
UNIT 1 NOTES | PAGE 31
HARTFIELD – PRECALCULUS
Solve the equation. Write the solution as a single
logarithmic expression and approximate the
solution to five places.
Ex. 8:
e2 x  3e x  2  0
UNIT 1 NOTES | PAGE 32
Solve the equation. Write the solution as a single
logarithmic expression and approximate the
solution to five places.
Ex. 9:
4 e2 x  e x  0
HARTFIELD – PRECALCULUS
Solve the equation. Check for extraneous
solutions.
Ex. 1:
log7 2   log7  x  1  log7  3x  5
UNIT 1 NOTES | PAGE 33
Solve the equation. Check for extraneous
solutions.
Ex. 2:
ln x  3  ln x  11  ln4  ln 4  x 
HARTFIELD – PRECALCULUS
Solve the equation. Check for extraneous
solutions.
Ex. 3:
log3  6 x   log3  x  4   2
UNIT 1 NOTES | PAGE 34
Solve the equation. Check for extraneous
solutions.
Ex. 4:
log4  x  8   log 4  x  4   3