Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

문제 53a

3장, 문제 53a

Find
a. (fog) (x)
b. (go f) (x)
c. (fog) (2)
d. (go f) (2).
f(x) = x+4, g(x) = 2x + 1

검증된 단계별 안내

Step 1: Understand the problem. You are tasked with finding the compositions of two functions, f(x) = x + 4 and g(x) = 2x + 1. Specifically, you need to compute (f ∘ g)(x), (g ∘ f)(x), (f ∘ g)(2), and (g ∘ f)(2). The notation (f ∘ g)(x) means f(g(x)), and (g ∘ f)(x) means g(f(x)).

Step 2: Compute (f ∘ g)(x). Substitute g(x) = 2x + 1 into f(x). This means replacing x in f(x) = x + 4 with g(x). The result is f(g(x)) = (2x + 1) + 4. Simplify the expression to get the formula for (f ∘ g)(x).

Step 3: Compute (g ∘ f)(x). Substitute f(x) = x + 4 into g(x). This means replacing x in g(x) = 2x + 1 with f(x). The result is g(f(x)) = 2(x + 4) + 1. Simplify the expression to get the formula for (g ∘ f)(x).

Step 4: Compute (f ∘ g)(2). Use the formula for (f ∘ g)(x) derived in Step 2 and substitute x = 2. Simplify the expression to find the value of (f ∘ g)(2).

Step 5: Compute (g ∘ f)(2). Use the formula for (g ∘ f)(x) derived in Step 3 and substitute x = 2. Simplify the expression to find the value of (g ∘ f)(2).

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)). This concept is essential for solving the given exercises, as it requires understanding how to manipulate and evaluate the functions in sequence.

Evaluating Functions

Evaluating functions means substituting a specific value into a function to find its output. For example, if f(x) = x + 4, then f(2) = 2 + 4 = 6. This skill is crucial for calculating the values of (fog)(2) and (go f)(2) in the exercises, as it requires substituting values into the composed functions.

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 algebraic manipulation is necessary for simplifying the results of the function compositions and ensuring accurate calculations in the exercises.

관련 실천

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교과서 질문

Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = x² - 2

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교과서 질문

In Exercises 53–58, f and g are defined by the following tables. Use the tables to evaluate each composite function. f(g(1))

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교과서 질문

In Exercises 55–59, use the graph of to graph each function g.

g(x) = f(x + 2) + 3

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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 + 2) (x = = −2, −1, 2, 7)

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교과서 질문

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Find a. (fog) (x) b. (go f) (x) c. (fog) (2) d. (go f) (2). f(x) = x+4, g(x) = 2x + 1

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) = x+4, g(x) = 2x + 1