교과서 질문
Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.
{Use of Tech} ƒ(x) = 1/(x-5)
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Ch. 1 - Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
1장, 문제 19e
Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.
Find the domain of ƒ o g.
검증된 단계별 안내1
insert step 1> Identify the functions involved in the composition. Here, we have \( f(x) = x^3 \) and \( g(x) = \sin x \).
insert step 2> Understand the composition \( (f \circ g)(x) = f(g(x)) \). This means we first apply \( g(x) \) and then apply \( f \) to the result.
insert step 3> Determine the domain of \( g(x) = \sin x \). The sine function is defined for all real numbers, so the domain of \( g \) is \( (-\infty, \infty) \).
insert step 4> Consider the domain of \( f(x) = x^3 \). The cube function is also defined for all real numbers, so the domain of \( f \) is \( (-\infty, \infty) \).
insert step 5> Since both \( g(x) \) and \( f(x) \) are defined for all real numbers, the domain of \( f \circ g \) is the same as the domain of \( g(x) \), which is \( (-\infty, \infty) \).
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Composite Functions
A composite function is formed when one function is applied to the result of another function. In mathematical notation, if f(x) and g(x) are two functions, the composite function f(g(x)) is evaluated by first calculating g(x) and then applying f to that result. Understanding how to combine functions is essential for solving problems involving their domains and ranges.
추천 영상:
3:48
Evaluate Composite Functions - Special Cases
Domain of a Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For composite functions, the domain is determined by the inner function and any restrictions imposed by the outer function. Identifying the domain is crucial for ensuring that all operations within the functions are valid and do not lead to undefined expressions.
추천 영상:
가이드 코스
5:10Finding the Domain and Range of a Graph
Restrictions on Functions
Restrictions on functions arise from operations that can limit the input values. For example, the square root function h(x) = √x is only defined for x ≥ 0, while the sine function g(x) = sin x is defined for all real numbers. When finding the domain of composite functions, it is important to consider these restrictions to ensure that the resulting function is valid across its entire domain.
추천 영상:
가이드 코스
7:24Multiplying & Dividing Functions
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Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.
Find the domain of g o ƒ.
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tan (15π/4)
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Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x .
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Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.
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