Find a. (fog) (x) b. (go f) (x) f(x) = 4-x, g(x) = 2x² +x+5

검증된 단계별 안내

Step 1: Understand the problem. You are tasked with finding two composite functions: (f ∘ g)(x) and (g ∘ f)(x). Composite functions involve substituting one function into another. Specifically, (f ∘ g)(x) means substituting g(x) into f(x), and (g ∘ f)(x) means substituting f(x) into g(x).

Step 2: Start with (f ∘ g)(x). Substitute g(x) = 2x² + x + 5 into f(x) = 4 - x. Replace 'x' in f(x) with the entire expression for g(x). This gives: f(g(x)) = 4 - (2x² + x + 5).

Step 3: Simplify the expression for (f ∘ g)(x). Distribute the negative sign across the terms inside the parentheses: f(g(x)) = 4 - 2x² - x - 5. Combine like terms to simplify further.

Step 4: Move on to (g ∘ f)(x). Substitute f(x) = 4 - x into g(x) = 2x² + x + 5. Replace 'x' in g(x) with the entire expression for f(x). This gives: g(f(x)) = 2(4 - x)² + (4 - x) + 5.

Step 5: Expand and simplify the expression for (g ∘ f)(x). First, expand (4 - x)² using the formula (a - b)² = a² - 2ab + b². Then distribute and combine like terms across the entire expression. This will yield the simplified form of g(f(x)).

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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)). Conversely, (go f)(x) means applying f first and then g, written as g(f(x)). Understanding this concept is crucial for solving the given problem.

Evaluating Functions

Evaluating functions requires substituting a specific input value into the function's formula to find the output. For example, if f(x) = 4 - x, to evaluate f(2), you would substitute 2 for x, resulting in f(2) = 4 - 2 = 2. This skill is essential for calculating the values of (fog)(x) and (go f)(x) in the exercise.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form g(x) = ax² + bx + c. In this case, g(x) = 2x² + x + 5 is a quadratic function. Understanding its properties, such as its graph's parabolic shape and how it interacts with linear functions, is important for analyzing the results of the compositions in the problem.