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.

f. y = √(x³ − 3)

검증된 단계별 안내

Identify the innermost function in the expression y = √(x³ − 3). Here, the innermost operation is x³, which corresponds to the function h(x) = x³.

Next, observe that after applying h(x), the expression becomes h(x) = x³. The next operation is subtraction by 3, which corresponds to the function f(x) = x − 3.

Combine these two operations: first apply h(x) to get x³, then apply f(x) to get x³ − 3. This can be expressed as f(h(x)).

Finally, the outermost operation is taking the square root, which corresponds to the function g(x) = √x. Apply g to the result of f(h(x)) to get g(f(h(x))).

Thus, the composition of functions that represents y = √(x³ − 3) is g(f(h(x))).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Composition of Functions

Composition of functions involves combining two or more functions to create a new function. If you have functions f(x) and g(x), the composition is denoted as (f ∘ g)(x) = f(g(x)). This means you apply g first and then apply f to the result. Understanding how to manipulate and combine functions is essential for solving problems that require expressing one function in terms of others.

Function Notation

Function notation is a way to represent functions and their outputs. For example, f(x) represents the output of function f when the input is x. This notation is crucial for understanding how to evaluate functions and perform operations like composition. Recognizing how to read and interpret function notation helps in identifying the relationships between different functions.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve equations or express functions in different forms. This skill is vital when working with compositions, as it allows you to substitute and combine functions effectively. Mastery of algebraic techniques, such as factoring, expanding, and simplifying, is necessary to express complex functions in terms of simpler ones.

Composition of FunctionsLet 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.f. y = √(x³ − 3)