Allied College Maryland Heights Even and Odd Functions and Symmetry Questions

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Question 1: (i) Determine if the following functions are even, odd, or neither. Show your algebraic justification foreach.(ii) Graph each functions using a graphing calculator or using an online graphing tool (such asDesmos.com). Provide a sketch or screenshot of your graphs and identify the symmetry for each.a) 𝑓(𝑥) = The equation is given in the screenshotb) 𝑔(𝑥) = The equation is given in the screenshotQuestion 2: Create.(a) Create your own piecewise-defined function to model a real-life situation such as your commute,a monthly salary (example #9, and problem #87 on page 212) or a cell phone plan (problem #88 page212). State the function’s equation and give a description of the function stating the input and outputof your function.(b) Sketch an accurate graph of your newly-created function using appropriate axes values. (Notethat in some cases it’s appropriate to use non-integer values for the independent variable.) Includevalues where the function changes.(c) Evaluate the piecewise function at three values and interpret what these points represent in thecontext of your problem.(d) Identify the domain and range appropriate for your problem situation. Write your answers usingset-builder notation.(e) State where the function is increasing, decreasing or constant using interval notation. (section 1.7page 205)

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SECTION 1.7
Analyzing Graphs of Functions and Piecewise-Defined Functions
SECTION 1.7Analyzing Graphs of Functions and Piecewise-Defined Functions
OBJECTIVES
1. Test for Symmetry
2. Identify Even and Odd Functions
3. Graph Piecewise-Defined Functions
4. Investigate Increasing, Decreasing, and Constant Behavior of a Function
5. Determine Relative Minima and Maxima of a Function
1. Test for Symmetry
The photos in Figures 1-28 through 1-30 each show a type of symmetry.
Figure 1-28
Figure 1-29
Lecture: Introduction to Symmetry
PDF Transcript for Lecture: Introduction to Symmetry
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Figure 1-30
The photo of the kingfisher (Figure 1-28) shows an image of the bird reflected in the water. Suppose
that we superimpose the x-axis at the waterline. Every point (x, y) on the bird has a mirror image (x,
−y) below the x-axis. Therefore, this image is symmetric with respect to the x-axis.
A human face is symmetric with respect to a vertical line through the center (Figure 1-29). If we
place the y-axis along this line, a point (x, y) on one side has a mirror image at (−x, y). This image is
symmetric with respect to the y-axis.
The flower shown in Figure 1-30 is symmetric with respect to the point at its center. Suppose that we
place the origin at the center of the flower. Notice that a point (x, y) on the image has a
corresponding point (−x, −y) on the image. This image is symmetric with respect to the origin.
Given an equation in the variables x and y, use the following rules to determine if the graph is
symmetric with respect to the x-axis, the y-axis, or the origin.
Tests for Symmetry
Consider an equation in the variables x and y.
The graph of the equation is symmetric with respect to the y-axis if substituting −x for x in the
equation results in an equivalent equation.
The graph of the equation is symmetric with respect to the x-axis if substituting −y for y in the
equation results in an equivalent equation.
The graph of the equation is symmetric with respect to the origin if substituting −x for x and −y
for y in the equation results in an equivalent equation.
Lecture: Testing for Symmetry
PDF Transcript for Lecture: Testing for Symmetry
Page 198
EXAMPLE 1Testing for Symmetry
Determine whether the graph is symmetric with respect to the y-axis, x-axis, or origin.
a. y = |x|
b. x = y2 − 4
Solution:
a.
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The graph is symmetric with respect to the y-axis only.
b.
The graph is symmetric with respect to the x-axis only (Figure 1-31).
Figure 1-31
TIP
The graph of y = |x| is one of the basic graphs presented in Section 1.6. From our familiarity with the
graph we can visualize the symmetry with respect to the y-axis.
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Skill Practice 1
Determine whether the graph is symmetric with respect to the y-axis, x-axis, or origin.
a. y = x2
b. |y| = x + 1
Answers
1.
a. y-axis
b. x-axis
EXAMPLE 2Testing for Symmetry
Determine whether the graph is symmetric with respect to the y-axis, x-axis, or origin.
x2 + y2 = 9
Solution:
The graph of x2 + y2 = 9 is a circle with center at the origin and radius 3. By inspection, we can see
that the graph is symmetric with respect to both axes and the origin.
Page 199
The graph is symmetric with respect to the y-axis, the x-axis, and the origin.
Skill Practice 2
Determine whether the graph is symmetric with respect to the y-axis, x-axis, or origin.
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Answer
2. y-axis, x-axis, and origin
2. Identify Even and Odd Functions
A function may be symmetric with respect to the y-axis or to the origin. A function that is symmetric
with respect to the y-axis is called an even function. A function that is symmetric with respect to the
origin is called an odd function.
Even and Odd Functions
A function f is an even function if f(−x) = f(x) for all x in the domain of f. The graph of an even
function is symmetric with respect to the y-axis.
A function f is an odd function if f(−x) = −f(x) for all x in the domain of f. The graph of an odd
function is symmetric with respect to the origin.
Avoiding Mistakes
The only functions that are symmetric with respect to the x-axis are functions whose points lie solely
on the x-axis.
Lecture: Introduction to Even and Odd Functions
PDF Transcript for Lecture: Introduction to Even and Odd Functions
EXAMPLE 3Identifying Even and Odd Functions
By inspection determine if the function is even, odd, or neither.
Solution:
a.
The function is symmetric with respect to the origin. Therefore, the function is an odd function.
b.
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The function is symmetric with respect to the y-axis. Therefore, the function is an even
function.
c.
The function is not symmetric with respect to either the y-axis or the origin. Therefore, the
function is neither even nor odd.
Page 200
Skill Practice 3
Determine if the function is even, odd, or neither.
a.
b.
c.
Answers
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a. Even function
b. Odd function
c. Neither even nor odd
Lecture: Determining Whether a Function is Even Odd or Neither
PDF Transcript for Lecture: Determining Whether a Function is Even Odd or Neither
EXAMPLE 4Identifying Even and Odd Functions
Determine if the function is even, odd, or neither.
a. f(x) = −2×4 + 5|x|
b. g(x) = 4×3 − x
c. h(x) = 2×2 + x
Solution:
a.
b.
Since g(−x) = −g(x), the function g is an odd function.
c.
Since
, the function is not even.
Next, test whether h is an odd function. Test whether h(−x) = −h(x).
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Since
, the function is not an odd function. Therefore, h is neither even nor odd.
TIP
In Example 4(a), we suspect that f is an even function because each term is of the form xeven or |x|.
In each case, replacing x by −x results in an equivalent term.
TIP
In Example 4(b), we suspect that g is an odd function because each term is of the form xodd. In each
case, replacing x by −x results in the opposite of the original term.
TIP
In Example 4(c), h(x) has a mixture of terms of the form xodd and xeven. Therefore, we might
suspect that the function is neither even nor odd.
Skill Practice 4
Determine if the function is even, odd, or neither.
a. m(x) = −x5 + x3
b. n(x) = x2 − |x| + 1
c. p(x) = 2|x| + x
Answers
4.
a. Odd function
b. Even function
c. Neither even nor odd
Page 201
3. Graph Piecewise-Defined Functions
Suppose that a car is stopped for a red light. When the light turns green, the car undergoes a
constant acceleration for 20 sec until it reaches a speed of 45 mph. It travels 45 mph for 1 min (60
sec), and then decelerates for 30 sec to stop at another red light. The graph of the car’s speed y (in
mph) versus the time x (in sec) after leaving the first red light is shown in Figure 1-32.
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Figure 1-32
Notice that the graph can be segmented into three pieces. The first 20 sec is represented by a linear
function with a positive slope, y = 2.25x. The next 60 sec is represented by the constant function y =
45. And the last 30 sec is represented by a linear function with a negative slope, y = −1.5x + 165.
To write a rule defining this function we use a piecewise-defined function in which we define each
“piece” on a restricted domain.
Lecture: Interpreting a Piecewise-Defined Function
PDF Transcript for Lecture: Interpreting a Piecewise-Defined Function
EXAMPLE 5Interpreting a Piecewise-Defined Function
Evaluate the function for the given values of x.
a. f(−3)
b. f(−1)
c. f(2)
d. f(6)
Solution:
a.
b.
c.
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d.
Skill Practice 5
Evaluate the function for the given values of x.
a. f(−3)
b. f(−2)
c. f(1)
d. f(4)
Answers
5.
a. 4
b. 4
c. 3
d. 3
Graphing Calculator: Graphing a Piecewise-Defined Function on a Calculator
Page 202
TECHNOLOGY CONNECTIONS
Graphing a Piecewise-Defined Function
A graphing calculator can be used to graph a piecewise-defined function. The format to enter the
function is as follows.
Each condition in parentheses is an inequality and the calculator assigns it a value of 1 or 0
depending on whether the inequality is true or false. If an inequality is true, the function is divided by
1 on that interval and is “turned on.” If an inequality is false, then the function is divided by 0. Since
division by zero is undefined, the calculator does not graph the function on that interval, and the
function is effectively “turned off.”
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Enter the function from Example 5 as shown. Note that the inequality symbols can be found in the
TEST menu.
Notice that the individual “pieces” of the graph do not “hook-up.” For this reason, it is also a good
practice to put the calculator in DOT mode in the
menu.
In Examples 6 and 7, we graph piecewise-defined functions.
Animation: Graphing a piecewise-defined function
Lecture: Graphing a Piecewise-Defined Function
PDF Transcript for Lecture: Graphing a Piecewise-Defined Function
EXAMPLE 6Graphing a Piecewise-Defined Function
Graph the function defined by
.
Solution:
The first rule f(x) = −3x defines a line with slope −3 and y-intercept (0, 0). This line should be
graphed only to the left of x = 1. The point (1, −3) is graphed as an open dot, because the
point is not part of the rule f(x) = −3x. See the blue portion of the graph in Figure 1-33.
Figure 1-33
The second rule f(x) = −3 is a horizontal line for x ≥ 1. The point (1, −3) is a closed dot to show
that it is part of the rule f(x) = −3. The closed dot from the red segment of the graph “overrides”
the open dot from the blue segment. Taken together, the closed dot “plugs” the hole in the
graph.
Page 203
Skill Practice 6
Graph the function.
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Answer
6.
TIP
The function in Example 6 has no “gaps,” and therefore we say that the function is continuous.
Informally, this means that we can draw the function without lifting our pencil from the page. The
formal definition of a continuous function will be studied in calculus.
EXAMPLE 7Graphing a Piecewise-Defined Function
Graph the function.
Solution:
The first rule f(x) = x + 3 defines a line with slope 1 and y-intercept (0, 3). This line should be
graphed only for x < −1 (that is to the left of x = −1). The point (−1, 2) is graphed as an open dot, because the point is not part of the function. See the red portion of the graph in Figure 1-34. Figure 1-34 The second rule f(x) = x2 is one of the basic functions learned in Section 1.6. It is a parabola with vertex at the origin. We sketch this function only for x values on the interval −1 ≤ x < 2. The point (−1, 1) is a closed dot to show that it is part of the function. The point (2, 4) is displayed as an open dot to indicate that it is not part of the function. Avoiding Mistakes Note that the function cannot have a closed dot at both (−1, 1) and (−1, 2) because it would not pass the vertical line test. TIP The function in Example 7 has a gap at x = −1, and therefore, we say that f is discontinuous at −1. https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 12/42 9/15/2020 ALEKS 360 Skill Practice 7 Graph the function. Answer 7. We now look at a special category of piecewise-defined functions called step functions. The graph of a step function is a series of discontinuous “steps.” One important step function is called the greatest integer function or floor function. It is defined by where is the greatest integer less than or equal to x. The operation may also be denoted as int(x) or by floor(x). These alternative notations are often used in computer programming. In Example 8, we graph the greatest integer function. Lecture: Introduction to the Greatest Integer Function PDF Transcript for Lecture: Introduction to the Greatest Integer Function Page 204 EXAMPLE 8Graphing the Greatest Integer Function Graph the function defined by . Solution: x Evaluate f for several values of x. −1.7 −2 Greatest integer less than or equal to −1.7 is −2. −1 −1 Greatest integer less than or equal to −1 is −1. −0.6 −1 Greatest integer less than or equal to −0.6 is −1. 0 0 Greatest integer less than or equal to 0 is 0. 0.4 0 Greatest integer less than or equal to 0.4 is 0. https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 13/42 9/15/2020 ALEKS 360 1 1 Greatest integer less than or equal to 1 is 1. 1.8 1 Greatest integer less than or equal to 1.8 is 1. 2 2 Greatest integer less than or equal to 2 is 2. 2.5 2 Greatest integer less than or equal to 2.5 is 2. TIP On many graphing calculators, the greatest integer function is denoted by int( ) and is found under the MATH menu followed by NUM. From the table, we see a pattern and from the pattern, we form the graph. Skill Practice 8 Evaluate f(x) = for the given values of x. a. f(1.7) b. f(5.5) c. f(−4) d. f(−4.2) Answers 8. a. 1 b. 5 c. −4 d. −5 In Example 9, we use a piecewise-defined function to model an application. EXAMPLE 9Using a Piecewise-Defined Function in an Application A salesperson makes a monthly salary of $3000 along with a 5% commission on sales over $20,000 for the month. Write a piecewise-defined function to represent the salesperson's monthly income I(x) (in $) for x dollars in sales. https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 14/42 9/15/2020 ALEKS 360 Solution: Let x represent the amount in sales. Then x − 20,000 represents the amount in sales over $20,000. There are two scenarios for the salesperson's income. Scenario The salesperson sells $20,000 or less. In this case, the monthly income is a constant 1: $3000. This is represented by Scenario The salesperson sells over $20,000. In this case, the monthly income is $3000 plus 5% of 2: sales over $20,000. This is represented by Page 205 Therefore, a piecewise-defined function for monthly income is Alternatively, we can simplify to get A graph of y = I(x) is shown in Figure 1-35. Notice that for x = $20,000, both equations within the piecewise-defined function yield a monthly salary of $3000. Therefore, the two line segments in the graph meet at (20,000, 3000). Figure 1-35 Skill Practice 9 A retail store buys T-shirts from the manufacturer. The cost is $7.99 per shirt for 1 to 100 shirts, inclusive. Then the price is decreased to $6.99 per shirt thereafter. Write a piecewise-defined function that expresses the cost C(x) (in $) to buy x shirts. Answer https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 15/42 9/15/2020 ALEKS 360 9. 4. Investigate Increasing, Decreasing, and Constant Behavior of a Function The graph in Figure 1-36 approximates the altitude of an airplane, f(t), at a time t minutes after takeoff. Figure 1-36 Notice that the plane's altitude increases up to the first 40 min of the flight. So we say that the function f is increasing on the interval (0, 40). The plane flies at a constant altitude for the next 1 hr 20 min, so we say that f is constant on the interval (40, 120). Finally, the plane's altitude decreases for the last 40 min, so we say that f is decreasing on the interval (120, 160). Informally, a function is increasing on an interval in its domain if its graph rises from left to right. A function is decreasing on an interval in its domain if the graph “falls” from left to right. A function is constant on an interval in its domain if its graph is horizontal over the interval. These ideas are stated formally using mathematical notation. Animation: Identifying the intervals of increasing decreasing and constant behavior on closed intervals Animation: Identifying the intervals of increasing decreasing and constant behavior on open intervals Lecture: Introduction to Increasing and Decreasing Functions PDF Transcript for Lecture: Introduction to Increasing and Decreasing Functions Page 206 Intervals of Increasing, Decreasing, and Constant Behavior Suppose that I is an interval contained within the domain of a function f. f is increasing on I if f(x1) < f(x2) for all x1 < x2 on I. f is decreasing on I if f(x1) > f(x2) for all x1 < x2 on I. f is constant on I if f(x1) = f(x2) for all x1 and x2 on I. https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 16/42 9/15/2020 ALEKS 360 Lecture: Determining the Intervals over Which a Function Increases Decreases or is Constant with Closed Intervals Lecture: Determining the Intervals over Which a Function Increases Decreases or is Constant with Open Intervals PDF Transcript for Lecture: Determining the Intervals over Which a Function Increases Decreases or is Constant with Closed Intervals PDF Transcript for Lecture: Determining the Intervals over Which a Function Increases Decreases or is Constant with Open Intervals EXAMPLE 10Determining the Intervals Over Which a Function Is Increasing, Decreasing, and Constant Use interval notation to write the interval(s) over which f is a. Increasing b. Decreasing c. Constant Solution: a. f is increasing on the interval (2, 3). (Highlighted in red tint.) b. f is decreasing on the interval (−4, −1) and (3, ∞). (Highlighted in orange tint.) c. f is constant on the interval (−1, 2). (Highlighted in green tint.) https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 17/42 9/15/2020 ALEKS 360 Skill Practice 10 Use interval notation to write the interval(s) over which f is a. Increasing b. Decreasing c. Constant Answers 10. a. (−5, −3) (−1, 2) b. (2, ∞) c. (−3, −1) Page 207 5. Determine Relative Minima and Maxima of a Function The intervals over which a function changes from increasing to decreasing behavior or vice versa tell us where to look for relative maximum values and relative minimum values of a function. Consider the function pictured in Figure 1-37. Figure 1-37 https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 18/42 9/15/2020 ALEKS 360 The function has a relative maximum of f(a). Informally, this means that f(a) is the greatest function value relative to other points on the function nearby. The function has a relative minimum of f(b). Informally, this means that f(b) is the smallest function value relative to other points on the function nearby. This is stated formally in the following definition. Relative Minimum and Relative Maximum Values f(a) is a relative maximum of f if there exists an open interval containing a such that f(a) ≥ f(x) for all x in the interval. f(b) is a relative minimum of f if there exists an open interval containing b such that f(b) ≤ f(x) for all x in the interval. Note: An open interval is an interval in which the endpoints are not included. TIP The plural of maximum and minimum are maxima and minima. Note that relative maxima and minima are also called local maxima and minima. If an ordered pair (a, f(a)) corresponds to a relative minimum or relative maximum, we interpret the coordinates of the ordered pair as follows. The x-coordinate is the location of the relative maximum or minimum within the domain of the function. The y-coordinate is the value of the relative maximum or minimum. This tells us how “high” or “low” the graph is at that point. Lecture: Introduction to Relative Maxima and Minima PDF Transcript for Lecture: Introduction to Relative Maxima and Minima EXAMPLE 11Finding Relative Maxima and Minima For the graph of y = g(x) shown, a. Determine the location and value of any relative maxima. b. Determine the location and value of any relative minima. Page 208 Solution: https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 19/42 9/15/2020 ALEKS 360 a. The point (2, 1) is the highest point in a small interval surrounding x = 2. Therefore, at x = 2, the function has a relative maximum of 1. b. The point (−1, −5) is the lowest point in a small interval surrounding x = −1. Therefore, at x = −1, the function has a relative minimum of −5. The point (4, −2) is the lowest point in a small interval surrounding x = 4. Therefore, at x = 4, the function has a relative minimum of −2. Avoiding Mistakes Be sure to note that the value of a relative minimum or relative maximum is the y value of a function, not the x value. Skill Practice 11 For the graph shown, a. Determine the location and value of any relative maxima. b. Determine the location and value of any relative minima. Answers 11. a. At x = −2, the function has a relative maximum of 3. b. At x = 2, the function has a relative minimum of 0. Graphing Calculator: Using a Graphing Calculator to Find the relative Minimum or Maximum of a Function TECHNOLOGY CONNECTIONS Determining Relative Maxima and Minima Relative maxima and relative minima are often difficult to find analytically and require techniques from calculus. However, a graphing utility can be used to approximate the location and value of relative maxima and minima. To do so, we use the Minimum and Maximum features. https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 20/42 9/15/2020 ALEKS 360 For example, enter the function defined by Y1 = x3 − 4x2 + 3x. Then access the Maximum feature from the CALC menu. The calculator asks for a left bound. This is a point slightly to the left of the relative maximum. Then hit ENTER. The calculator asks for a right bound. This is a point slightly to the right of the relative maximum. Hit ENTER. Page 209 The calculator asks for a guess. This is a point close to the relative maximum. Hit ENTER and the approximate coordinates of the relative maximum point are shown (0.45, 0.63). To find the relative minimum, repeat these steps using the Minimum feature. The coordinates of the relative minimum point are approximately (2.22, −2.11). SECTION 1.7Practice Exercises Prerequisite Review https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 21/42 9/15/2020 R.1. Given the function defined by ALEKS 360 , find f(–a). Answers For Exercises R.2–R.4, graph the set and express the set in interval notation. R.2. Answers R.3. Answers R.4. Answers Concept Connections 1. A graph of an equation is symmetric with respect to the results in an equivalent equation. -axis if replacing x by −x Answers 2. A graph of an equation is symmetric with respect to the results in an equivalent equation. -axis if replacing y by −y 3. A graph of an equation is symmetric with respect to the y by −y results in an equivalent equation. if replacing x by −x and Answers 4. An even function is symmetric with respect to the 5. An odd function is symmetric with respect to the . . Answers 6. The expression represents the greatest integer, less than or equal to x. Objective 1: Test for Symmetry For Exercises 7–18, determine whether the graph of the equation is symmetric with respect to the x-axis, y-axis, origin, or none of these. (See Examples 1–2) 7. y = x2 + 3 Answers 8. y = −|x| − 4 9. x = −|y| − 4 Answers https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 22/42 9/15/2020 ALEKS 360 10. x = y2 + 3 Exercise: Testing for Symmetry PDF Transcript for Exercise: Testing for Symmetry 11. x2 + y2 = 3 Answers 12. |x| + |y| = 4 Exercise: Testing for Symmetry PDF Transcript for Exercise: Testing for Symmetry 13. y = |x| + 2x + 7 Answers 14. y = x2 + 6x + 1 15. x2 = 5 + y2 Answers 16. y4 = 2 + x2 17. Answers 18. Objective 2: Identify Even and Odd Functions 19. What type of symmetry does an even function have? Answers 20. What type of symmetry does an odd function have? Page 210 For Exercises 21–26, use the graph to determine if the function is even, odd, or neither. (See Example 3) 21. https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 23/42 9/15/2020 22. ALEKS 360 Answers 23. 24. Answers 25. 26. Answers https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 24/42 9/15/2020 27. ALEKS 360 a. Given f(x) = 4x2 − 3|x|, find f(−x). b. Is f(−x) = f(x)? c. Is this function even, odd, or neither? Answers 28. a. Given g(x) = −x8 + |3x|, find g(−x). b. Is g(−x) = g(x)? c. Is this function even, odd, or neither? 29. a. Given h(x) = 4x3 − 2x, find h(−x). b. Find −h(x). c. Is h(−x) = −h(x)? d. Is this function even, odd, or neither? Answers 30. a. Given k(x) = −8x5 − 6x3, find k(−x). b. Find −k(x). c. Is k(−x) = −k(x)? d. Is this function even, odd, or neither? 31. a. Given m(x) = 4x2 + 2x − 3, find m(−x). b. Find −m(x). c. Is m(−x) = m(x)? d. Is m(−x) = −m(x)? e. Is this function even, odd, or neither? Answers 32. a. Given n(x) = 7|x| + 3x − 1, find n(−x). b. Find −n(x). https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 25/42 9/15/2020 ALEKS 360 c. Is n(−x) = n(x)? d. Is n(−x) = −n(x)? e. Is this function even, odd, or neither? For Exercises 33–46, determine if the function is even, odd, or neither. (See Example 4) 33. Answers 34. Exercise: Determining Whether a Function is Even Odd or Neither PDF Transcript for Exercise: Determining Whether a Function is Even Odd or Neither 35. Answers 36. 37. Answers 38. 39. Exercise: Determining Whether a Function is Even Odd or Neither PDF Transcript for Exercise: Determining Whether a Function is Even Odd or Neither Answers 40. 41. Answers 42. 43. Answers 44. 45. https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 26/42 9/15/2020 ALEKS 360 Answers 46. Objective 3: Graph Piecewise-Defined Functions For Exercises 47–50, evaluate the function for the given values of x. (See Example 5) 47. a. f(3) b. f(−2) c. f(−1) d. f(4) e. f(5) Exercise: Interpreting a Piecewise-Defined Function PDF Transcript for Exercise: Interpreting a Piecewise-Defined Function Answers 48. a. g(−3) b. g(3) c. g(−2) d. g(0) e. g(4) Page 211 49. a. h(−1.7) b. h(−2.5) c. h(0.05) https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 27/42 9/15/2020 ALEKS 360 d. h(−2) e. h(0) Answers 50. a. t(1.99) b. t(0.4) c. t(3) d. t(1) e. t(3.001) 51. A sled accelerates down a hill and then slows down after it reaches a flat portion of ground. The speed of the sled s(t) (in ft/sec) at a time t (in sec) after movement begins can be approximated by: Determine the speed of the sled after 10 sec, 20 sec, 30 sec, and 40 sec. Round to 1 decimal place if necessary. Answers 52. A car starts from rest and accelerates to a speed of 60 mph in 12 sec. It travels 60 mph for 1 min and then decelerates for 20 sec until it comes to rest. The speed of the car s(t) (in mph) at a time t (in sec) after the car begins motion can be modeled by: Determine the speed of the car 6 sec, 12 sec, 45 sec, and 80 sec after the car begins motion. For Exercises 53–56, match the function with its graph. 53. f(x) = x + 1 for x < 2 Answers 54. f(x) = x + 1 for −1 < x ≤ 2 https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 28/42 9/15/2020 ALEKS 360 55. f(x) = x + 1 for −1 ≤ x < 2 Answers 56. f(x) = x + 1 for x ≥ 2 a. b. c. d. 57. a. Graph p(x) = x + 2 for x ≤ 0. (See Examples 6–7) b. Graph q(x) = −x2 for x > 0.
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a. Graph f(x) = |x| for x < 0. b. c. 59. a. b. c. Answers 60. a. Graph a(x) = x for x < 1. b. c. For Exercises 61–70, graph the function. (See Examples 6–7) 61. Answers 62. 63. Answers 64. 65. Answers 66. Page 212 https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 30/42 9/15/2020 ALEKS 360 67. Answers 68. Exercise: Graphing a Piecewise-Defined Function PDF Transcript for Exercise: Graphing a Piecewise-Defined Function 69. Answers 70. 71. a. b. To what basic function from Section 1.6 is the graph of f equivalent? Answers For Exercises 72–80, evaluate the step function defined by (See Example 8) for the given values of x. 72. f(−3.7) 73. f(−4.2) Answers 74. f(−0.5) 75. f(−0.09) Answers 76. f(0.5) 77. f(0.09) Answers 78. f(6) 79. f(−9) https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 31/42 9/15/2020 ALEKS 360 Answers 80. f(−5) For Exercises 81–84, graph the function. (See Example 8) 81. Answers 82. 83. Answers 84. h(x) = int(2x) 85. For a recent year, the rate for first class postage was as follows. (See Example 9) Weight not OverPrice 1 oz $0.44 2 oz $0.61 3 oz $0.78 3.5 oz $0.95 Write a piecewise-defined function to model the cost C(x) to mail a letter first class if the letter is x ounces. Answers 86. The water level in a retention pond started at 5 ft (60 in.) and decreased at a rate of 2 in./day during a 14-day drought. A tropical depression moved through at the beginning of the 15th day and produced rain at an average rate of 2.5 in./day for 5 days. Write a piecewise-defined function to model the water level L(x) (in inches) as a function of the number of days x since the beginning of the drought. 87. A salesperson makes a base salary of $2000 per month. Once he reaches $40,000 in total sales, he earns an additional 5% commission on the amount in sales over $40,000. Write a piecewise-defined function to model the salesperson's total monthly salary S(x) (in $) as a function of the amount in sales x. Answers 88. A cell phone plan charges $49.95 per month plus $14.02 in taxes, plus $0.40 per minute for calls beyond the 600-min monthly limit. Write a piecewise-defined function to model the https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 32/42 9/15/2020 ALEKS 360 monthly cost C(x) (in $) as a function of the number of minutes used x for the month. Objective 4: Investigate Increasing, Decreasing, and Constant Behavior of a Function For Exercises 89–96, use interval notation to write the intervals over which f is (a) increasing, (b) decreasing, and (c) constant. (See Example 10) 89. Exercise: Determining the Intervals over Which a Function Increases Decreases or Constant with Closed Intervals Exercise: Determining the Intervals over Which a Function Increases Decreases or Constant with Open Intervals PDF Transcript for Exercise: Determining the Intervals over Which a Function Increases Decreases or Constant with Closed Intervals PDF Transcript for Exercise: Determining the Intervals over Which a Function Increases Decreases or Constant with Open Intervals 90. Answers 91. 92. Answers https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 33/42 9/15/2020 ALEKS 360 93. 94. 95. 96. Answers Page 213 Answers Objective 5: Determine Relative Minima and Maxima of a Function https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 34/42 9/15/2020 ALEKS 360 For Exercises 97–102, identify the location and value of any relative maxima or minima of the function. (See Example 11) 97. 98. Answers Exercise: Introduction to Relative Maxima and Minima PDF Transcript for Exercise: Introduction to Relative Maxima and Minima 99. 100. Answers 101. https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 35/42 9/15/2020 102. ALEKS 360 Answers 103. The graph shows the depth d (in ft) of a retention pond, t days after recording began. a. Over what interval(s) does the depth increase? b. Over what interval(s) does the depth decrease? c. Estimate the times and values of any relative maxima or minima on the interval (0, 20). d. If rain is the only water that enters the pond, explain what the intervals of increasing and decreasing behavior mean in the context of this problem. Answers 104. The graph shows the height h (in meters) of a roller coaster t seconds after the ride starts. https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 36/42 9/15/2020 ALEKS 360 a. Over what interval(s) does the height increase? b. Over what interval(s) does the height decrease? c. Estimate the times and values of any relative maxima or minima on the interval (0, 70). Page 214 Mixed Exercises For Exercises 105–110, produce a rule for the function whose graph is shown. (Hint: Consider using the basic functions learned in Section 1.6 and transformations of their graphs.) 105. 106. Answers 107. https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 37/42 9/15/2020 108. ALEKS 360 Answers 109. 110. Answers For Exercises 111–112, a. Graph the function. b. Write the domain in interval notation. c. Write the range in interval notation. d. Evaluate f(−1), f(1), and f(2). e. Find the value(s) of x for which f(x) = 6. f. Find the value(s) of x for which f(x) = −3. https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 38/42 9/15/2020 ALEKS 360 g. Use interval notation to write the intervals over which f is increasing, decreasing, or constant. 111. Answers 112. In computer programming, the greatest integer function is sometimes called the “floor” function. Programmers also make use of the “ceiling” function, which returns the smallest integer not less than x. For example: ceil(3.1) = 4. For Exercises 113–114, evaluate the floor and ceiling functions for the given value of x. floor(x) is the greatest integer less than or equal to x. ceil(x) is the smallest integer not less than x. 113. a. floor(2.8) b. floor(−3.1) c. floor(4) d. ceil(2.8) e. ceil(−3.1) f. ceil(4) Answers 114. a. floor(5.5) b. floor(−0.1) c. floor(−2) d. ceil(5.5) e. ceil(−0.1) f. ceil(−2) Write About It 115. From an equation in x and y, explain how to determine whether the graph of the equation is symmetric with respect to the x-axis, y-axis, or origin. Answers 116. From the graph of a function, how can you determine if the function is even or odd? 117. Explain why the relation defined by https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 39/42 9/15/2020 ALEKS 360 is not a function. Answers 118. Explain why the function is discontinuous at x = 1. Page 215 119. Provide an informal explanation of a relative maximum. Answers 120. Explain what it means for a function to be increasing on an interval. Expanding Your Skills 121. Suppose that the average rate of change of a continuous function between any two points to the left of x = a is negative, and the average rate of change of the function between any two points to the right of x = a is positive. Does the function have a relative minimum or maximum at a? Answers 122. Suppose that the average rate of change of a continuous function between any two points to the left of x = a is positive, and the average rate of change of the function between any two points to the right of x = a is negative. Does the function have a relative minimum or maximum at a? A graph is concave up on a given interval if it “bends” upward. A graph is concave down on a given interval if it “bends” downward. For Exercises 123–126, determine whether the curve is (a) concave up or concave down and (b) increasing or decreasing. 123. 124. Answers 125. https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 40/42 9/15/2020 126. ALEKS 360 Answers 127. For a recent year, the federal income tax owed by a taxpayer (single—no dependents) was based on the individual's taxable income. (Source: Internal Revenue Service, www.irs.gov) If your taxable income is over— but not over—The tax is of the amount over— $0 $0 $8925 $0 + 10% $8925 $36,250 $892.50 + 15% $8925 $36,250$87,850 $4991.25 + 25%$36,250 Write a piecewise-defined function that expresses an individual's federal income tax f(x) (in $) as a function of the individual's taxable income x (in $). Answers Technology Connections For Exercises 128–131, use a graphing utility to graph the piecewise-defined function. 128. 129. Exercise: Graphing a Piecewise-Defined Function on a Calculator Answers 130. 131. For Exercises 132–135, use a graphing utility to https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 41/42 9/15/2020 ALEKS 360 a. Find the locations and values of the relative maxima and relative minima of the function on the standard viewing window. Round to 3 decimal places. b. Use interval notation to write the intervals over which f is increasing or decreasing. Answers 132. 133. Answers 134. 135. Answers https://www-awu.aleks.com/alekscgi/x/Isl.exe/1o_u-IgNsIkr7j8P3jH-lBhBKpIvJLUgSfgdbAAD_YYslMyIlKt4xb8yPQ0B6p5CDtYEc1k1Q5tj2TUNY_S_… 42/42 Purchase answer to see full attachment Tags: functions even ODD neither algebraic justification User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.