Let ƒ(x)=x2+3 and g(x)=-2x+6. Find each of the following. (ƒg)(4)

Ch. 2 - Graphs and Functions

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Ch. 2 - Graphs and Functions

문제 15

3장, 문제 15

Let ƒ(x)=x²+3 and g(x)=-2x+6. Find each of the following. (ƒg)(4)

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Understand that (ƒg)(4) means the composition of functions ƒ and g evaluated at 4, which is written as ƒ(g(4)). This means you first find g(4), then substitute that result into ƒ(x).

Calculate g(4) by substituting 4 into the function g(x) = -2x + 6. So, compute g(4) = -2(4) + 6.

Simplify the expression for g(4) to find its value. This will give you the input for the next step.

Substitute the value of g(4) into the function ƒ(x) = x^2 + 3. This means you replace x in ƒ(x) with the value you found for g(4).

Simplify the expression ƒ(g(4)) by squaring the value from g(4) and then adding 3, which will give you the final expression for (ƒg)(4).

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주요 개념

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

Function Composition

Function composition involves applying one function to the result of another, denoted as (f∘g)(x) = f(g(x)). It means you first evaluate g at x, then use that output as the input for f. This concept is essential for understanding how to combine functions and evaluate them at specific values.

Evaluating Functions at a Given Input

Evaluating a function at a given input means substituting the input value into the function's formula and simplifying to find the output. For example, to find g(4), replace x with 4 in g(x) = -2x + 6 and simplify. This step is crucial before composing functions.

Polynomial and Linear Functions

Understanding the forms of polynomial functions like f(x) = x² + 3 and linear functions like g(x) = -2x + 6 helps in correctly substituting and simplifying expressions. Recognizing these types aids in performing arithmetic operations and function evaluations accurately.