Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

문제 51

3장, 문제 51

Find
a. (fog) (x)
b. (go f) (x)
c. (fog) (2)
d. (go f) (2).
f(x) = 2x, g(x) = x+7

검증된 단계별 안내

Step 1: Understand the problem. You are tasked with finding the compositions of two functions, f(x) = 2x and g(x) = x + 7. Specifically, you need to compute (f ∘ g)(x), (g ∘ f)(x), (f ∘ g)(2), and (g ∘ f)(2). The symbol ∘ represents function composition, meaning you substitute one function into the other.

Step 2: To find (f ∘ g)(x), substitute g(x) into f(x). This means replacing the input of f(x) with g(x). Start with f(x) = 2x, and replace x with g(x) = x + 7. The result is f(g(x)) = 2(x + 7). Simplify this expression.

Step 3: To find (g ∘ f)(x), substitute f(x) into g(x). This means replacing the input of g(x) with f(x). Start with g(x) = x + 7, and replace x with f(x) = 2x. The result is g(f(x)) = 2x + 7. Simplify this expression.

Step 4: To find (f ∘ g)(2), use the expression for (f ∘ g)(x) that you derived in Step 2. Substitute x = 2 into the expression f(g(x)) = 2(x + 7). Simplify the result.

Step 5: To find (g ∘ f)(2), use the expression for (g ∘ f)(x) that you derived in Step 3. Substitute x = 2 into the expression g(f(x)) = 2x + 7. Simplify the result.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Function Composition

Function composition involves combining two functions to create a new function. If f(x) and g(x) are two functions, the composition (fog)(x) means applying g first and then f to the result, expressed as f(g(x)). Understanding this concept is crucial for solving problems that require evaluating composite functions.

Evaluating Functions

Evaluating functions means substituting a specific value into a function to find its output. For example, if f(x) = 2x, then f(2) = 2(2) = 4. This skill is essential for calculating the values of composite functions at given points, as required in the question.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying expressions using algebraic rules. This includes operations like addition, multiplication, and applying the distributive property. Mastery of these techniques is necessary to simplify the results of function compositions and to evaluate them accurately.

관련 실천

교과서 질문

In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)

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교과서 질문

Graph using intercepts: 2x - 5y - 10 = 0

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교과서 질문

In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = √x (x = 0, 1, 4, 9) and g(x) = √x −1 (x = 0, 1, 4, 9)

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교과서 질문

Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. (x + 1)² + y² = 25

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교과서 질문

Use the graph of y = f(x) to graph each function g. g(x) =(1/2) f(2x)

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교과서 질문

In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = ∛x + 1

586

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Find a. (fog) (x) b. (go f) (x) c. (fog) (2) d. (go f) (2). f(x) = 2x, g(x) = x+7

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Function Composition

Evaluating Functions

Algebraic Manipulation

Find
a. (fog) (x)
b. (go f) (x)
c. (fog) (2)
d. (go f) (2).
f(x) = 2x, g(x) = x+7