copyright © cengage learning. all rights reserved. 1.3 graphs of functions
TRANSCRIPT
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Copyright © Cengage Learning. All rights reserved.
1.3 Graphs of Functions
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What You Should Learn
• Find the domains and ranges of functions anduse the Vertical Line Test for functions
• Determine intervals on which functions are increasing, decreasing, or constant
• Determine relative maximum and relative minimum values of functions
• Identify and graph piecewise-defined functions
• Identify even and odd functions
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The Graph of a Function
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Example 1 – Finding the Domain and Range of a Function
Use the graph of the function f shown below to find:
(a) the domain of f, (b) the function values f (–1) and f (2), and (c) the range of f.
Figure 1.18
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Example 1 – Solution
a. The closed dot at (–1, –5) indicates that x = –1 is in the domain of f, whereas the open dot at (4, 0) indicates
that x = 4 is not in the domain. So, the domain of f is all x in the interval [–1, 4).
Therefore, Domain = [-1,4) which includes -1, but not 4.
b. Because (–1, –5) is a point on the graph of f, it follows that
f (–1) = –5.
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Example 1 – Solution
Similarly, because (2, 4) is a point on the graph of f, it follows that
f (2) = 4.
c. Because the graph does not extend below f (–1) = –5
or above f (2) = 4, the range of is the interval [–5, 4].
Therefore, Range = [-5,4] which includes -5 and 4.
cont’d
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The Graph of a Function
By the definition of a function, each x value may only correspond to one y value. It follows, then, that avertical line can intersect the graph of a function at mostonce. This leads to the Vertical Line Test for functions.
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Example 3 – Vertical Line Test for Functions
Use the Vertical Line Test to decide whether the graphs in Figure 1.19 represent y as a function of x.
Figure 1.19
(a) (b)
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Example 3 – Solution
a. This is not a graph of y as a function of x because youcan find a vertical line that intersects the graph twice.
b. This is a graph of y as a function of x because every vertical line intersects the graph at most once.
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Increasing and Decreasing Functions
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Increasing and Decreasing FunctionsConsider the graph shown below.
Moving from left to right, this graph falls from x = –2 to x = 0, is constant from x = 0 to x = 2, and rises from x = 2 to x = 4.
Figure 1.20
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Increasing and Decreasing Functions
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Example 4 – Increasing and Decreasing Functions
In Figure 1.21, determine the open intervals on which each function is increasing, decreasing, or constant.
Figure 1.21
(a) (b) (c)
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Example 4 – Solution
a. Although it might appear that there is an interval in which this function is constant, you can see thatif x1 < x2, then (x1)3 < (x2)3, which implies thatf (x1) < f (x2).
This means that the cube of a larger number is bigger than the cube of a smaller number. So, the function is increasing over the entire real line.
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Example 4 – Solution
b. This function is increasing on the interval ( , –1),
decreasing on the interval (–1, 1), and increasing on the
interval (1, ).
c. This function is increasing on the interval ( , 0), constant on the interval (0, 2), and decreasing on the interval (2, ).
cont’d
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Relative Minimum and Maximum Values
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Relative Minimum and Maximum Values
The points at which a function changes its increasing, decreasing, or constant behavior are helpful in determiningthe relative maximum or relative minimum values of thefunction.
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Relative Minimum and Maximum ValuesFigure 1.22 shows several different examples of relative minima and relative maxima.
Figure 1.22
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Piecewise-Defined Functions
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Example 8 – Sketching a Piecewise-Defined Function
Sketch the graph of
2x + 3, x ≤ 1
–x + 4, x > 1
by hand.
f (x) =
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Example 8 – Solution
Solution:
This piecewise-defined function is composed of two linear
functions.
At and to the left of x = 1, the graph is the line
given by
y = 2x + 3.
To the right of x = 1, the graph is the line given by
y = –x + 4
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Example 8 – Solution
The two linear functions are combined and below.
Notice that the point (1, 5) is a solid dot and the point (1, 3) is an open dot. This is because f (1) = 5.
Figure 1.29
cont’d
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Even and Odd Functions
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Even and Odd Functions
A graph has symmetry with respect to the y-axis ifwhenever (x, y) is on the graph, then so is the point (–x, y).
A graph has symmetry with respect to the origin if whenever (x, y) is on the graph, then so is the point (–x, –y).
A graph has symmetry with respect to the x-axis if whenever (x, y) is on the graph, then so is the point (x, –y).
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Even and Odd Functions
A function whose graph is symmetric with respect to the y -axis is an even function.
A function whose graph is symmetric with respect to the origin is an odd function.
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Even and Odd Functions
A graph that is symmetric with respect to the x-axis is notthe graph of a function (except for the graph of y = 0). These three types of symmetry are illustrated in Figure 1.30.
Symmetric to y-axisEven function
Symmetric to originOdd function
Symmetric to x-axisNot a function
Figure 1.30
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Even and Odd Functions
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Example 10 – Even and Odd Functions
Determine whether each function is even, odd, or neither.
a. g(x) = x3 – x
b. h(x) = x2 + 1
c. f (x) = x3 – 1
Solution:
a. This function is odd because
g (–x) = (–x)3+ (–x)
= –x3 + x
= –(x3 – x)
= –g(x).
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Example 10 – Solution
b. This function is even because
h (–x) = (–x)2 + 1
= x2 + 1
= h (x).
c. Substituting –x for x produces
f (–x) = (–x)3 – 1
= –x3 – 1.
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Example 10 – Solution
Because
f (x) = x3 – 1
and
–f (x) = –x3 + 1
you can conclude that
f (–x) f (x)
and
f (–x) –f (x).
So, the function is neither even nor odd.
cont’d