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Section 1.2
Functions and
Graphs
Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
Objectives
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Determine whether a correspondence or a relation is a
function.
Find function values, or outputs, using a formula or a
graph.
Graph functions.
Determine whether a graph is that of a function.
Find the domain and the range of a function.
Solve applied problems using functions.
Function
A function is a correspondence between a first set, called
the domain, and a second set, called the range, such
that each member of the domain corresponds to exactly
one member of the range.
It is important to note that not every correspondence
between two sets is a function.
Example
Determine whether each of the following
correspondences is a function.
a.
This correspondence is a function
6
because each member of the
36
6
domain corresponds to exactly
3
one member of the range. The
9
definition allows more than one
3
member of the domain to
0
0
correspond to the same member
of the range.
Example
Determine whether each of the following
correspondences is a function.
b.
The Queen
Helen Mirren
Blood Diamond
Jennifer Hudson
Leonardo DiCaprio
Dreamgirls
Jamie Foxx
The Departed
This correspondence is not a function because there
is one member of the domain (Leonardo DiCaprio)
that is paired with more than one member of the
range (Blood Diamond and The Departed).
Relation
A relation is a correspondence between the first set,
called the domain, and a second set, called the range,
such that each member of the domain corresponds to at
least one member of the range.
Example
Determine whether each of the following relations is
a function. Identify the domain and range.
a. {(9, 5), (9, 5), (2, 4)}
Not a function. Ordered
pairs (9, –5) and (9, 5) have
the same first coordinate
and different second
coordinates.
Domain is the set of first coordinates: {9, 2}.
Range is the set of second coordinates: {–5, 5, 4}.
Example (continued)
Determine whether each of the following relations is
a function. Identify the domain and range.
b. {(–2, 5), (5, 7), (0, 1), (4, –2)}
Is a function. No two ordered
pairs have the same first
coordinate and different
second coordinates.
Domain is the set of first coordinates: {–2, 5, 0, 4}.
Range is the set of second coordinates: {5, 7, 1, –2}.
Example (continued)
Determine whether each of the following relations is a
function. Identify the domain and range.
b. {(–5, 3), (0, 3), (6, 3)}
Is a function. No two ordered
pairs have the same first
coordinate and different
second coordinates.
Domain is the set of first coordinates: {–5, 0, 6}.
Range is the set of second coordinates: {3}.
Notation for Functions
The inputs (members of the domain) are values of x
substituted into the equation. The outputs (members of
the range) are the resulting values of y.
f (x) is read “f of x,” or “f at x,” or “the value of f at x.”
Example
A function is given by f(x) = 2x2  x + 3. Find each of the
following.
a. f (0)
f (0) = 2(0)2  0 + 3 = 0 – 0 + 3 = 3
b. f (–7)
f (–7) = 2(–7)2  (–7) + 3 = 2 • 49 + 7 + 3 = 108
Example (continued)
A function is given by f(x) = 2x2  x + 3. Find each of the
following.
c. f (5a)
f (5a) = 2(5a)2  5a + 3 = 2 • 25a2 – 5a + 3
= 50a2 – 5a + 3
d. f (a – 4)
f (a – 4) = 2(a – 4)2  (a – 4) + 3
= 2(a2 – 8a + 32) – a + 4 + 3
= 2a2 – 16a + 64 – a + 4 + 3
= 2a2 – 17a + 71
Graphs of Functions
We graph functions the same way we graph equations.
We find ordered pairs (x, y), or (x, f (x)), plot the points
and complete the graph.
Example
Graph f (x) = x2 – 5 .
Make a table of values.
x
f (x)
4
3
–1
2
–1
–4
0
5
1
4
2
3
1
4
(x, f (x))
(3, 4)
(2, –1)
(–1, –4)
(0, 5)
(1, 4)
(2, 1)
(3, 4)
Example (continued)
Graph f (x) = x3 – x .
Example (continued)
Graph f x  
x4.
Example
For the function f (x) = x2 – 5, use the graph to find each
of the following function values.
a. f (3)
b. f (–2)
a. Locate the input 3 on
the horizontal axis,
move vertically (up) to
the graph of the
function, then move
horizontally (left) to
find the output on the
vertical axis.
f (3) = 4
Example (cont)
For the function f (x) = x2 – 5, use the graph to find each
of the following function values.
a. f (3)
b. f (–2)
b. Locate the input –2 on
the horizontal axis,
move vertically (down)
to the graph, then
move horizontally
(right) to find the output
on the vertical axis.
f (–2) = –1
Vertical-Line Test
If it is possible for a vertical line to cross a graph
more than once, then the graph is not the graph of
a function.
Example
Which of graphs (a) - (c) (in red) are graphs of functions?
Yes.
No.
No.
Example (continued)
Which of graphs (d) - (f) (in red) are graphs of functions?
In graph (f), the solid dot shows that (–1, 1) belongs to
the graph. The open circle shows that (–1, –2) does not
belong to the graph.
No.
Yes.
Yes.
Finding Domains of Functions
When a function f whose inputs and outputs are real
numbers is given by a formula, the domain is understood
to be the set of all inputs for which the expression is
defined as a real number. When an input results in an
expression that is not defined as a real number, we say
that the function value does not exist and that the number
being substituted is not in the domain of the function.
Example
Find the indicated function values and determine whether
the given values are in the domain of the function.
1
f (x) 
x3
a. f (1)
1
1
f (1) 

1 3
2
Since f (1) is defined, 1 is in the domain of f.
Copyright © 2009 Pearson Education, Inc.
Example (cont)
Find the indicated function values and determine whether
the given values are in the domain of the function.
1
f (x) 
x3
b. f (3)
1
1
f (3) 

3 3 0
Since division by 0 is not defined, f (3) does not exist
and, 3 is in not in the domain of f.
Copyright © 2009 Pearson Education, Inc.
Example
2
3x
x7
Find the domain of the function h(x) 
.
2
x  2x  3
We can substitute any real number in the numerator,
but we must avoid inputs that make the denominator
0. Solve x2 + 2x  3 = 0.
(x + 3)(x – 1) = 0
x + 3 = 0 or x – 1 = 0
x = –3 or
x=1
The domain consists of the set of all real numbers
except 3 and 1, or {x|x  3 and x  1}.
Visualizing Domain and Range
Keep the following in mind regarding the graph of a
function:
Domain = the set of a function’s inputs, found on the
horizontal axis;
Range = the set of a function’s outputs, found on the
vertical axis.
Example
Graph the function. Then estimate the domain and range.
f ( x) 
x4
Domain = [–4, )
Range = [0, )