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Functions and
Their Graphs
OBJECTIVE :
• Identify and graph linear and squaring
functions.
•
Recognize EVEN and ODD functions
• Identify and graph cubic, square root, and
reciprocal function.
• Identify and graph step and other
piecewise-defined functions.
• Recognize graphs of parent functions.
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Linear and Squaring Functions
The graph of the linear function y= mx+b
• The domain of the function is the set of all real numbers.
• The range of the function is the set of all real numbers.
• The graph has an x-intercept of (–b/m, 0) and
a y-intercept of (0, b).
• The graph is increasing if slope m  0, and decreasing if
slope m  0, and constant (horizontal line) if m = 0.
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Example 1 – Writing a Linear Function
Write the linear function for which f (1) = 3 and
f (4) = 0.
Solution:
To find the equation of the line that passes through
(x1, y1) = (1, 3) and (x2, y2) = (4, 0) first find the slope
of the line.
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Example 1 – Solution
cont’d
Next, use the point-slope form of the equation of a line.
y – y1 = m(x – x1)
y – 3 = –1(x – 1)
y = –x + 4
f(x) = –x + 4
Point-slope form
Substitute for x1, y1 and m
Simplify.
Function notation
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Example 1 – Solution
cont’d
The graph of this function is shown in Figure 1.65.
Figure 1.65
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Linear and Squaring Functions
There are two special types of linear functions, the
constant function and the identity function.
A constant function has the form
f(x) = c
and has the domain of all real numbers with a range
consisting of a single real number c.
The graph of a constant function is
a horizontal line, as shown in
Figure 1.66. The identity function
has the form f(x) = x.
Figure 1.66
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Linear and Squaring Functions
Its domain and range are the set of all real numbers. The
identity function has a slope of m = 1 and a y-intercept at
(0, 0).
The graph of the identity function is a line for which each
x-coordinate equals the corresponding y-coordinate. The
graph is always increasing, as shown in Figure 1.67.
Figure 1.67
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Linear and Squaring Functions
The graph of the squaring function
f (x) = x2
is a U-shaped curve with the following characteristics.
• The domain of the function - all real numbers.
• The range - all nonnegative
• The function is even :
real numbers.
f(-x) = f(x)
• The graph has an intercept at (0, 0).
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Linear and Squaring Functions
• The graph is decreasing on the interval (
increasing on the interval (0, )
, 0) and
• The graph is symmetric with respect to the y-axis.
• The graph has a relative minimum at (0, 0).
Figure 1.68
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Cubic, Square Root, and Reciprocal Functions
1. Cubic function
f (x) = x3
• The
domain - all real numbers.
• The
range - all real numbers.
• The
function is odd : f(-x)
f(-x) =
3
(-x)
=-
3
x
= - f(x)
= - f(x)
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The graph of the cubic function
• The graph is increasing on the interval (
,
).
• The graph is symmetric with respect to the origin.
Cubic function
Figure 1.69
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Square Root Function
2. The graph of the square root function f(x) =
• The domain – all nonnegative
• The range - all nonnegative
real numbers.
real numbers.
• The graph has an intercept at (0, 0).
• The Graph is increasing on the interval (0,
)
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The Graph of Square Root Function
Square root function
Figure 1.70
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Reciprocal Function
3. The graph of the reciprocal function f(x) = has the
following characteristics.
• The domain of the function is(
, 0)  (0, )
• The range of the function is (
, 0)  (0,
)
• The function is odd : f(-x) = - f(x)
• The graph does not have any y or x intercepts.
• The graph is decreasing on the intervals
(
, 0) and (0, ).
• The graph is symmetric with respect to the origin.
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The graph of the Reciprocal Functions
.
Reciprocal function
Figure 1.71
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Step and Piecewise-Defined Functions
Step functions: Functions whose graphs resemble sets of
stair steps
The most famous of the step functions is the
greatest integer function, which is denoted by
defined as
f(x) =
and
= the greatest integer less than or equal to x.
Some values of the greatest integer function are as follows.
= (greatest integer  –1) = –1
= (greatest integer 
) = –1
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Step and Piecewise-Defined Functions
= (greatest integer 
)=0
= (greatest integer  1.5) = 1
The graph of the greatest integer function
f(x) =
• The domain of the STEP function - all real numbers.
• The range of the STEP function - all integers.
• The graph has a y-intercept at (0, 0) and x-intercepts in
the interval [0, 1).
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Step Function GRAPH
Figure 1.72
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Example 2 – Evaluating a Step Function
Evaluate the function when x = –1, 2 and
f(x) =
+1
Solution:
For x = –1, the greatest integer  –1 is –1, so
f(–1) =
+1
= –1 + 1
=0
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Example 2 – Solution
cont’d
For x = 2, the greatest integer  2 is 2, so
f(2) =
+1
=2+1
= 3.
For x =
, the greatest integer 
is 1, so
=1+1
=2
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Example 2 – Solution
cont’d
You can verify your answers by examining the graph of
f(x) =
+ 1 shown in Figure 1.73.
Figure 1.73
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Parent Functions
The most commonly used functions in algebra.
(a) Constant Function
(b) Identity Function
(c) Absolute Value
Function
(d) Square Root
Function
Figure 1.75
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Parent Functions
(e) Quadratic
Function
(f) Cubic
Function
(g) Reciprocal
Function
Figure 1.75
(h) Greatest
Integer
Function
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