Textbook Question
Solving Trigonometric Equations
For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.
sin² θ = cos² θ
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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 1.1.52
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
g(x) = x⁴ + 3x² − 1
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 g(x) is even if g(-x) = g(x) for all x in the domain, and it is odd if g(-x) = -g(x) for all x in the domain.
Start by substituting -x into the function g(x) = x⁴ + 3x² − 1 to find g(-x). This gives us g(-x) = (-x)⁴ + 3(-x)² − 1.
Simplify the expression for g(-x). Since (-x)⁴ = x⁴ and (-x)² = x², we have g(-x) = x⁴ + 3x² − 1.
Compare g(-x) with g(x). We find that g(-x) = x⁴ + 3x² − 1 is exactly the same as g(x) = x⁴ + 3x² − 1, which means g(-x) = g(x).
Since g(-x) = g(x), the function g(x) is even. Therefore, g(x) = x⁴ + 3x² − 1 is an even 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 classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain. This means that the graph of an even function is symmetric with respect to the y-axis. Common examples include polynomial functions with only even powers of x, such as x² or x⁴.
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Exponential Functions
Odd Functions
A function is considered odd if it meets the condition f(-x) = -f(x) for all x in its domain. This indicates that the graph of an odd function is symmetric with respect to the origin. Typical examples include polynomial functions with only odd powers of x, like x³ or x⁵.
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Neither Even Nor Odd Functions
A function is classified as neither even nor odd if it does not satisfy the conditions for either classification. This can occur when a function contains both even and odd powers of x or does not exhibit symmetry about the y-axis or the origin. Analyzing the function's behavior at -x compared to x helps determine this classification.
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Related Practice
Textbook Question
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = sec x tan x
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Textbook Question
Functions and Graphs
Find the natural domain and graph the functions in Exercises 15–20.
f(x) = 1 − 2x − x²
386
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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.
sin x²
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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.
g(x) = x/(x² − 1)
206
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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 = −x³
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