Textbook Question
Graph the function y = √|x|.
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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 1.16
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x cos x
Verified step by step guidance1
To determine if a function is even, odd, or neither, we need to analyze the function's symmetry properties. A function f(x) is even if f(-x) = f(x) for all x in the domain, and it is odd if f(-x) = -f(x) for all x in the domain.
Start by substituting -x into the function y = x cos(x). This gives us y(-x) = (-x) cos(-x).
Recall that the cosine function is even, meaning cos(-x) = cos(x). Therefore, y(-x) = (-x) cos(x).
Now, compare y(-x) = (-x) cos(x) with the original function y = x cos(x). Notice that y(-x) = -y(x), which satisfies the condition for the function to be odd.
Since y(-x) = -y(x), the function y = x cos(x) is classified as an odd function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Even Functions
A function is considered even if it satisfies the condition f(-x) = f(x) for all x in its domain. This means that the graph of the function is symmetric with respect to the y-axis. For example, the function f(x) = x² is even because f(-x) = (-x)² = x².
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Odd Functions
A function is classified as odd if it meets the condition f(-x) = -f(x) for all x in its domain. This indicates that the graph of the function is symmetric with respect to the origin. An example of an odd function is f(x) = x³, as f(-x) = (-x)³ = -x³.
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Function Analysis
Function analysis involves evaluating the properties of a function, such as its symmetry, continuity, and limits. In the context of determining if a function is even, odd, or neither, one typically substitutes -x into the function and compares the result to f(x) and -f(x). This analysis is crucial for understanding the behavior of functions in calculus.
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Related Practice
Textbook Question
Algebraic Combinations
In Exercises 3 and 4, find the domains of f, g, f/g and g/f.
f(x) = 1, g(x) = 1 + √x
Textbook Question
In Exercises 39–42, express the given quantity in terms of sin x and cos x.
cos (3π/2 + x)
Textbook Question
Finding Formulas for Functions
Consider the point (x,y) lying on the graph of y = √(x − 3). Let L be the distance between the points (x,y) and (4,0). Write L as a function of y.
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Textbook Question
Finding Formulas for Functions
Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length.
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Textbook Question
General Sine Curves
For
f(x) = A sin ((2π/B)(x – C) +D
identify A, B, C, and D for the sine functions in Exercises 67–70 and sketch their graphs.
y = ½ sin (πx – x) + ½
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