교과서 질문
Solving Trigonometric Equations
For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.
sin² θ = cos² θ
293
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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
1장, 문제 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
검증된 단계별 안내1
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.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
6:13
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.
추천 영상:
06:21Properties of Functions
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.
추천 영상:
06:21Properties of Functions
관련 실천
교과서 질문
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = sec x tan x
236
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교과서 질문
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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교과서 질문
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
sin x²
246
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교과서 질문
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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교과서 질문
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
280
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