```Functions
and Their Graphs
Chapter 3
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.
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.
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.
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.
Vertical Line Test
Evaluate
f  x  1 given that f  x   x  3 x .
2
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.
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
 Understand that odd functions
have graphs that are symmetric
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
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
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.
Section 3.3
Graphing Techniques: Transformations
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
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
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
Vertical Stretching and Vertical
Compressing of Graphs
Horizontal Stretching and Horizontal
Compressing of Graphs
Skills Objectives
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.
Section 3.4
Operations on Functions
and Composition of Functions
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
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
Section 3.5
One-to-One Functions
and Inverse Functions
Horizontal Line Test
Inverse Functions
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