Let ƒ(x)=x2+3ƒ(x)=$$x^2+3 $$and g(x)=−2x+6g(x)=-2x+6. Find each of the following. See Example 1.(ƒg)(−3) (ƒg)(-3)

Ch. 2 - Graphs and Functions

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Lial 13th Edition

Ch. 2 - Graphs and Functions

문제 16

3장, 문제 16

Let $ƒ (x) = x^{2} + 3$ and $g (x) = - 2 x + 6$ . Find each of the following. See Example 1.
$(ƒ g) (- 3)$

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Understand that (ƒg)(x) means the composition of functions ƒ and g, which is ƒ(g(x)). This means you first apply g to x, then apply ƒ to the result.

Start by finding g(-3) by substituting -3 into the function g(x) = -2x + 6. Write the expression: $g(-3) = -2(-3) + 6$.

Simplify the expression for g(-3) to find the value that will be input into ƒ. This involves multiplying and adding the numbers inside the parentheses.

Next, substitute the value found for g(-3) into the function ƒ(x) = x^2 + 3. Write the expression: $ƒ(g(-3)) = (g(-3))^2 + 3$.

Finally, simplify the expression by squaring the value of g(-3) and then adding 3 to find the value of (ƒg)(-3).

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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 (ƒg)(x) = ƒ(g(x)). It means you first evaluate g at x, then use that output as the input for ƒ. 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(-3), replace x with -3 in g(x) = -2x + 6 and simplify. This step is crucial before composing functions.

Polynomial and Linear Functions

ƒ(x) = x² + 3 is a polynomial function, and g(x) = -2x + 6 is a linear function. Understanding their forms helps in correctly substituting and simplifying expressions during evaluation and composition. Recognizing these types aids in anticipating the behavior and complexity of the functions.

Let ƒ(x)=x2+3ƒ(x)=x^2+3 and g(x)=−2x+6g(x)=-2x+6. Find each of the following. See Example 1.(ƒg)(−3) (ƒg)(-3)