교과서 질문
The accompanying figure shows the graph of a function f(x) with domain [0,2] and range [0,1]. Find the domains and ranges of the following functions, and sketch their graphs.
<IMAGE>
c. 2f(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
1장, 문제 1.2.16c
Composition of Functions
Evaluate each expression using the functions
f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0
x − 1, 0 ≤ x ≤ 2
c. g(g(−1))
검증된 단계별 안내1
First, identify the function g(x) that applies to the input value x = -1. Since -1 falls within the interval -2 ≤ x < 0, use the expression g(x) = -x.
Evaluate g(-1) using the expression g(x) = -x. Substitute -1 for x, which gives g(-1) = -(-1).
Simplify the expression -(-1) to find the value of g(-1).
Now, use the result from g(-1) as the new input for the function g(x) again. Determine which expression of g(x) applies to this new input value.
Evaluate g(g(-1)) using the appropriate expression for g(x) based on the new input value obtained from the previous step.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Composition of Functions
Composition of functions involves combining two functions where the output of one function becomes the input of another. For example, if you have functions f(x) and g(x), the composition g(f(x)) means you first apply f to x, then apply g to the result. Understanding this concept is crucial for evaluating expressions like g(g(-1)).
추천 영상:
3:48
Evaluate Composite Functions - Special Cases
Piecewise Functions
A piecewise function is defined by different expressions based on the input value. In this case, g(x) has two different rules depending on the value of x: one for -2 ≤ x < 0 and another for 0 ≤ x ≤ 2. Recognizing which part of the function to use based on the input is essential for correctly evaluating g(-1) and subsequently g(g(-1)).
추천 영상:
05:36Piecewise Functions
Function Evaluation
Function evaluation is the process of finding the output of a function for a given input. This involves substituting the input value into the function's formula. For instance, to evaluate g(-1), you need to determine which piece of the piecewise function applies and then compute the result accordingly, which is a key step in solving the problem.
추천 영상:
4:26Evaluating Composed Functions
관련 실천
교과서 질문
Composition of Functions
Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.
c. y = x¹/⁴
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교과서 질문
Combining Functions
Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?
d. f² = ff
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교과서 질문
Composition of Functions
Copy and complete the following table.
b. <IMAGE>
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교과서 질문
Functions
In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.
b. <IMAGE>
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교과서 질문
Find the largest interval on which the given function is increasing.
c. g(x) = (3x - 1)¹/³
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