Let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. (fog) (0)

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Identify the functions involved: \(f(x) = 2x - 5\) and \(g(x) = 4x - 1\). The problem asks for \((f \circ g)(0)\), which means \(f(g(0))\).

First, evaluate the inner function \(g(0)\) by substituting \(x = 0\) into \(g(x) = 4x - 1\). This gives \(g(0) = 4(0) - 1\).

Simplify the expression for \(g(0)\) to find its value.

Next, take the result from \(g(0)\) and substitute it into the outer function \(f(x) = 2x - 5\). So, compute \(f(g(0)) = 2 \times g(0) - 5\).

Simplify the expression for \(f(g(0))\) to find the final value of \((f \circ g)(0)\).

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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 the inner function g at x, then use that output as the input for f. Understanding this process is essential to correctly evaluate composite functions without finding their explicit formula.

Evaluating Functions at a Specific Input

Evaluating a function at a specific input means substituting the given value into the function's formula and simplifying. For composite functions, this requires careful step-by-step substitution, first evaluating the inner function at the input, then using that result in the outer function.

Order of Operations in Function Evaluation

When evaluating composite functions, the order of operations is crucial: first compute the inner function's value, then apply the outer function to that result. This ensures accurate evaluation and avoids errors that arise from mixing up the sequence of substitutions.