교과서 질문
Piecewise-Defined Functions
Find a formula for each function graphed in Exercises 29–32.
b. <IMAGE>
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
1장, 문제 1.1.8b
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>
검증된 단계별 안내1
Step 1: Understand the definition of a function. A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
Step 2: Use the Vertical Line Test to determine if a graph represents a function. The Vertical Line Test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.
Step 3: Examine the given graph (represented by ) and imagine drawing vertical lines across the entire graph.
Step 4: Check if any vertical line intersects the graph at more than one point. If it does, the graph is not a function of x. If every vertical line intersects the graph at most once, then it is a function of x.
Step 5: Provide a reasoned conclusion based on the Vertical Line Test. If the graph passes the test, state that it is a function of x. If it fails, explain that it is not a function of x because there exists at least one vertical line that intersects the graph at more than one point.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Definition of a Function
A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. This means that for every x-value in the domain, there is a unique y-value in the range. Understanding this definition is crucial for determining whether a given graph represents a function.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral
Vertical Line Test
The vertical line test is a method used to determine if a graph represents a function. If any vertical line drawn through the graph intersects it at more than one point, the graph does not represent a function. This test provides a visual way to assess the uniqueness of outputs for each input.
추천 영상:
가이드 코스
05:13Slopes of Tangent Lines
Domain and Range
The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). Understanding the domain and range helps in analyzing the behavior of functions and is essential for identifying whether a graph meets the criteria of a function.
추천 영상:
가이드 코스
5:10Finding the Domain and Range of a Graph
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
Composition of Functions
Copy and complete the following table.
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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교과서 질문
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))
203
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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>
369
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