Find a. (fog) (x) b. the domain of f o g.
f(x) = x² + 4, g(x) = √(1 − x)

검증된 단계별 안내

Step 1: Understand the composition of functions. The notation (fog)(x) represents the composition of f and g, meaning f(g(x)). To find this, substitute g(x) into f(x).

Step 2: Substitute g(x) = √(1 − x) into f(x) = x² + 4. This gives f(g(x)) = (√(1 − x))² + 4.

Step 3: Simplify the expression. Since (√(1 − x))² simplifies to 1 − x, the composition becomes f(g(x)) = 1 − x + 4.

Step 4: Combine like terms to simplify further. The final expression for (fog)(x) is f(g(x)) = 5 − x.

Step 5: Determine the domain of f o g. The domain of g(x) is restricted by the square root, so 1 − x ≥ 0, which simplifies to x ≤ 1. Additionally, f(g(x)) must be defined, and since there are no further restrictions in f(g(x)), the domain of f o g is x ≤ 1.

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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, where the output of one function becomes the input of another. In this case, (f o g)(x) means applying g first and then applying f to the result of g. Understanding how to correctly substitute and evaluate these functions is crucial for solving the problem.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the composition of functions, the domain of f o g is determined by the domain of g and the values that g outputs that are also valid inputs for f. This requires analyzing both functions to ensure all inputs are permissible.

Square Root Function

The square root function, denoted as g(x) = √(1 - x), is defined only for non-negative values under the square root. This means that the expression 1 - x must be greater than or equal to zero, which imposes restrictions on the domain of g. Understanding these restrictions is essential for determining the overall domain of the composite function f o g.

Find a. (fog) (x) b. the domain of f o g. f(x) = x² + 4, g(x) = √(1 − x)