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When working with functions, the first step is evaluating them. This means calculating the output value for a given input, using the function's formula. For example, if we have a function \( g(x) = 2 - x^2 \), and we want to evaluate it at \( x = -2 \), we substitute \(-2\) for \( x \) in the equation. This gives us:\[ g(-2) = 2 - (-2)^2 = 2 - 4 = -2 \]In this case, evaluating the function tells us that when \( x = -2 \), \( g(x) \) is \(-2\). Evaluation is a fundamental step in working with any function, as it helps us understand how the function behaves with different inputs.

Substitution in functions involves replacing the variable in a function's equation with a specific numerical value. This is crucial when we are working with operations that require specific inputs. Once you substitute the given value, you can perform the necessary arithmetic to obtain a clear result.Consider the function \( f(x) = 3x - 5 \). To find \( f(-2) \), substitute \(-2\) into the function:\[ f(-2) = 3(-2) - 5 = -6 - 5 = -11 \]By substituting \(-2\) for \(x\), we calculate that the output is \(-11\). This process of substitution is critical in determining individual outputs that will later be used in compositions.

Operations with function inputs often involve compositions, where one function's output becomes another function's input. This process can extend functions' capabilities by combining their actions.Function composition is typically written as \((f \circ g)(x)\), which means you apply \(g(x)\) and use the output as the input for \(f(x)\). For example, to solve \((f \circ g)(-2)\), start by finding \(g(-2)\):\[ g(-2) = 2 - (-2)^2 = -2 \]Then use this output for the function \(f(x)\):\[ f(g(-2)) = f(-2) = 3(-2) - 5 = -11 \]In this case, the final result for the composition of \(f\) and \(g\) at \(-2\) is \(-11\). Having a good understanding of these operations helps in tackling complex function interactions efficiently.