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Chapter 7
Functions and
Graphs
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7.2
Domain and Range
• Determining the Domain and the Range
• Restrictions on Domain
• Piecewise-Defined Functions
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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.
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Example
Find the domain and range of the
function f below.
6
f
5
4
Here f can be written
{(–5, 1), (1, 0), (3, –5), (4, 3)}.
3
2
The domain is the set of all first
coordinates, {–5, 1, 3, 4}.
1
-5 -4 -3 -2 -1-1
-2
-3
1
2
3 4
5
The range is the set of all
second coordinates, {1, 0, –5, 3}.
-4
-5
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Example
For the function f represented below, determine
each of the following.
y
a) What member of the
range is paired with -2
7
6
5
4
b) The domain of f
c) What member of the
domain is paired with 6
f
3
2
1
-5 -4 -3 -2 -1
-1
d) The range of f
1
2 3 4 5
x
-2
-3
-4
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a) What member of the range is paired with -2
Solution
y
Locate -2 on the horizontal axis
(this is where the domain is
located). Next, find the point
directly above -2 on the graph of
f. From that point, look to the
corresponding y-coordinate, 3.
The “input” -2 has the “output” 3.
7
6
5
f
4
3
2
Output
1
-5 -4 -3 -2 -1
-1
Input -2
1
2 3 4 5
x
-3
-4
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b) The domain of f
Solution
y
The domain of f is the set
of all x-values that are used
in the points on the curve.
These extend continuously
from −5 to 3 and can be
viewed as the curve’s
shadow, or projection, on
the x-axis. Thus the
domain is {x | 5  x  3}.
7
6
5
f
4
3
2
1
-5 -4 -3 -2 -1
-1
1
2 3 4 5
The domain
of f
-2
x
-3
-4
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c) What member of the domain is paired with 6
Solution
y
Locate 6 on the vertical axis
(this is where the range is
located). Next, find the point
to the right of 6 on the graph
of f. From that point, look to
the corresponding xcoordinate, 2.5. The “output”
6 has the “input” 2.5.24
7
Output
6
5
f
4
3
2
1
-5 -4 -3 -2 -1
-1
-2
1
2 3 4 5
Input
x
-3
-4
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d) The range of f
Solution
The range of f is the set
of all y-values that are
used in the points on the
curve. These extend
continuously from -1 to 7
and can be viewed as the
curve’s shadow, or
projection, on the y-axis.
Thus the range is
y
7
The
range of
f
6
5
f
4
3
2
1
-5 -4 -3 -2 -1
-1
{ y | 1  y  7}.
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2 3 4 5
x
-2
-3
-4
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Example
Determine the domain of f ( x)  3x2  4.
Solution
We ask, “Is there any number x for
which we cannot compute 3x2 – 4?”
Since the answer is no, the domain of
f is the set of all real numbers.
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2
Example Determine the domain of f ( x) 
.
x 8
Solution
We ask, “Is there any number x for which
2
cannot be computed?” Since 2 cannot
x 8
x 8
be computed when x – 8 = 0 the answer is
yes.
To determine what x-value would cause x – 8 to
x – 8 = 0,
be 0, we solve an equation:
x=8
Thus 8 is not in the domain of f, whereas all
other real numbers are. The domain of f is
{x | x is a real number and x  8}.
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Piecewise-Defined Functions
Some functions are defined by different
equations for various parts of their
domains. Such functions are said to be
piecewise-defined. For example, the function
given by f(x) = |x| can be described by
 x, if x  0
f (x)  
 x, if x  0
To evaluate a piecewise-defined function for an
input a, we determine what part of the domain a
belongs to and use the appropriate formula for
that part of the domain.
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Example
Find each function value for the function f given by
3 x, if x  4
f (x)  
 x  2, if x  4
a. f(5)
b. f(–8)
Solution
a. Determine which equation to use.
5 is in the second part of the domain 5  4
f (x)  x  2
f (5)  5  2  7
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Example
Find each function value for the function f given by
3 x, if x  4
f (x)  
 x  2, if x  4
a. f(5)
b. f(–8)
Solution
b. Determine which equation to use.
–8 is in the first part of the domain 8  4
f ( x )  3x
f ( 8)  3( 8)  24
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Example
Find each function value for the function f given by
 x  3, if x  3
 2
f ( x)   x ,
if  3  x  4
 4 x,
if x  4

a) f(3)
b) f(2)
c) f(7)
Solution
a) f(x) = x + 3: f(3) = 3 + 3 = 0
b)f(x) = x2; f(2) = 22 = 4
c)f(7) = 4x = 4(7) = 28
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