Functions
and Their Graphs
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Chapter 3
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Chapter 3
Overview
 Find the domain and range of a function.
 Sketch the graphs of common functions.
 Sketch graphs of general functions
employing translations of common
functions.
 Perform composition of functions.
 Find the inverse of a function.
 Model applications with functions using
variation.
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Chapter 3
Objectives
Skills Objectives
 Determine whether a
relation is a function.
 Determine whether an
equation represents a
function.
 Use function notation.
 Find the value of a
function.
 Determine the domain
and range of a function.
Conceptual Objectives
 Think of function notation
as a placeholder or
mapping.
 Understand that all
functions are relations but
not all relations are
functions.
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Section 3.1
Functions
A function is a
correspondence between
two sets where each
element in the first set
corresponds exactly to one
element in the second set.
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Function
Given a graph of an equation, if any vertical
line that can be drawn intersects the graph
at no more than one point, the equation
defines y as a function of x. This test is
called the vertical line test.
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Vertical Line Test
Evaluate
f  x  1 given that f  x   x  3 x .
2
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Common Mistake
F x  
State the domain of the given function.
3
x
2
 25
Solution :
3
F(x) 
Write the orginial equation.
x
Determine
any restrictio ns on the values of x .
Solve the restrictio n equation.
State the domain
Write domain
The domain is restricted
 25
 25  0
2
 25 or x   25   5
x  5
   ,  5    5 , 5   5 ,  
notation.
to all real numbers
2
x
restrictio ns.
in interval
x
2
except
 5 and 5.
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Domain of a Function
Graphs of Functions; Piecewise-Defined Functions;
Increasing and Decreasing Functions; Average Rate of
Change
Skills Objectives
Conceptual Objectives
 Classify functions as even, odd,
or neither.
 Determine whether functions are
increasing, decreasing, or
constant.
 Calculate the average rate of
change of a function.
 Evaluate the difference quotient
for a function.
 Graph piecewise-defined
functions.
 Identify common functions.
 Develop and graph piecewisedefined functions:
 Identify and graph points of
discontinuity.
 State the domain and range.
 Understand that even functions
have graphs that are symmetric
about the y-axis.
 Understand that odd functions
have graphs that are symmetric
about the origin.
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Section 3.2
Graph the piecewise-defined function, and state the intervals
where the function is increasing, decreasing, or constant, along
with the domain and range.
x  1
 x

f x    2
 x

1 x  1
x 1
Click mouse to continue
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Your Turn!
Graph the piecewise-defined function, and state the intervals
where the function is increasing, decreasing, or constant, along
with the domain and range.
x  1
 x

f x    2
 x

Increasing
Decreasing
Constant
: 1,  
:    ,  1
:   1, 1
Domain :    , 1  1,  
Range : 1,  
1 x  1
x 1
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Your Turn!
Skills Objectives
Conceptual Objectives
 Sketch the graph of a function
using horizontal and vertical
shifting of common functions.
 Sketch the graph of a function
by reflecting a common
function about the x-axis or yaxis.
 Sketch the graph of a function
by stretching or compressing a
common function.
 Sketch the graph of a function
using a sequence of
transformations.
 Identify the common
functions by their graphs.
 Apply multiple
transformations of common
functions to obtain graphs of
functions.
 Understand that domain and
range are also transformed.
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Section 3.3
Graphing Techniques: Transformations
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Vertical and
Horizontal Shifts
The graph of –f(x) is obtained by reflecting
the function f (x) about the x-axis.
The graph of f(-x) is obtained by rotating the
function f(x) about the y-axis.
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Reflection About the Axes
Use shifts and reflection
State the domain
to graph the
function
f x   
and range of f  x 
Click mouse to continue
x 1  2
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Your Turn!
Use shifts and reflection
State the domain
to graph the
and range of f  x 
Domain : 1,  
Range : -  , 2 
function
f x   
x 1  2
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Your Turn!
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Vertical Stretching and Vertical
Compressing of Graphs
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Horizontal Stretching and Horizontal
Compressing of Graphs
Skills Objectives
 Add, subtract, multiply,
and divide functions.
 Evaluate composite
functions.
 Determine domain of
functions resulting from
operations and
composition of functions.
Conceptual Objectives
 Understand domain
restrictions when dividing
functions.
 Realize that the domain of a
composition of functions
excludes the values that are
not in the domain of the
inside function.
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Section 3.4
Operations on Functions
and Composition of Functions
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Composition of Functions
Given the functions
Evaluate
f x   x
2
 7 and g  x   5  x
2
f g 1
Solution:
One way of evaluating these composite functions is to calculate the
two individual composites in terms of x: f(g(x)) and g(f(x)). Once those
functions are known, the values can be substituted for x and
evaluated. Another way of proceeding is as follows:
f g 1
Write the desired quantity.
Find the value of the inner function,
Subsitute
g.
g 1  4 into f .
g 1  5  1  4
2
f g 1  f  4 
Evalue f  4 .
f 4   4  7  9
2
f g 1  9
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Evaluating a
Composite Function
Skills Objectives
 Determine whether a
function is a one-to-one
function.
 Verify that two functions
are inverses of one
another.
 Graph the inverse
function given the graph
of the function.
 Find the inverse of a
function.
Conceptual Objectives
 Visualize the relationships
between the domain and
range of a function and the
domain and range of its
inverse.
 Understand why functions
and their inverses are
symmetric about y = x.
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Section 3.5
One-to-One Functions
and Inverse Functions
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Horizontal Line Test
College Algebra, Third Edition by Cynthia Y. Young, © 2012 John Wiley and Sons. All rights reserved.
Inverse Functions
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