Textbook Question
Use graphing software to graph the functions specified in Exercises 31–36.
Select a viewing window that reveals the key features of the function.
Graph the function f (x) = sin 2x + cos 3x.
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.8
In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.
𝔂 = e⁻ˣ²
Verified step by step guidance1
To determine symmetry about the y-axis, check if the function y = f(x) satisfies f(x) = f(-x). Substitute -x into the function: y = e^(-(-x)^2). Simplify to see if it equals the original function.
To determine symmetry about the origin, check if the function y = f(x) satisfies f(-x) = -f(x). Substitute -x into the function: y = e^(-(-x)^2) and compare it to -e^(-x^2).
Simplify the expression e^(-(-x)^2) to e^(-x^2) and compare it to the original function e^(-x^2) to check for y-axis symmetry.
Compare e^(-x^2) with -e^(-x^2) to check for origin symmetry. If they are not equal, the function is not symmetric about the origin.
Conclude whether the function is symmetric about the y-axis, the origin, or neither based on the comparisons made in the previous steps.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Symmetry about the y-axis
A function is symmetric about the y-axis if replacing x with -x in the function yields the same output. Mathematically, this means that f(-x) = f(x) for all x in the domain of the function. This type of symmetry indicates that the graph is a mirror image across the y-axis.
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06:21
Properties of Functions
Symmetry about the origin
A function is symmetric about the origin if replacing x with -x and y with -y results in the same equation. This is expressed as f(-x) = -f(x). Functions with this symmetry exhibit rotational symmetry of 180 degrees around the origin, meaning that if you rotate the graph, it looks the same.
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06:21Properties of Functions
Exponential functions
Exponential functions, such as y = e^(-x²), are characterized by a constant base raised to a variable exponent. These functions typically exhibit rapid growth or decay. Understanding their general shape and behavior is crucial for analyzing their symmetry properties, as they often do not possess symmetry about the y-axis or the origin.
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6:13Exponential Functions
Related Practice
Textbook Question
Algebraic Combinations
In Exercises 1 and 2, find the domains of f, g, f + g, and f ⋅ g.
f(x) = √(x + 1), g(x) = √(x − 1)
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Textbook Question
Increasing and Decreasing Functions
Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
y = (−x)²/³
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Textbook Question
[Technology Exercise]
a. Graph the functions f(x) = x/2 and g(x) = 1 + (4/x) together to identify the values of x for which
x/2 > 1 + 4/x
b. Confirm your findings in part (a) algebraically.
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Textbook Question
Finding a Viewing Window
In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.
y = x + (1/10) sin 30x
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Textbook Question
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
f(x) = x² + x
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