# 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
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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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.
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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
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)
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
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
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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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
c.
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