Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. (g∘ƒ)(0)

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Understand that the notation \((g \circ f)(0)\) means the composition of the functions \(g\) and \(f\) evaluated at \(x=0\). This is written as \(g(f(0))\), which means you first find \(f(0)\) and then plug that result into \(g\).

Calculate \(f(0)\) by substituting \(x=0\) into the function \(f(x) = 2x - 3\). So, \(f(0) = 2(0) - 3\).

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

Take the result from \(f(0)\) and substitute it into the function \(g(x) = -x + 3\). This means you replace \(x\) in \(g(x)\) with the value you found for \(f(0)\), so you compute \(g(f(0)) = -[f(0)] + 3\).

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

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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 (g∘ƒ)(x) = g(ƒ(x)). It means you first evaluate ƒ at x, then use that output as the input for g. This process combines two functions into a single operation.

Evaluating Functions at a Given Input

Evaluating a function at a specific input means substituting the input value into the function's formula and simplifying to find the output. For example, ƒ(0) means replacing x with 0 in ƒ(x) and calculating the result.

Linear Functions

Linear functions have the form f(x) = mx + b, where m and b are constants. They produce straight-line graphs and are easy to evaluate and compose because their outputs change at a constant rate with respect to x.