Ch. 2 - Functions and Graphs

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Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problema 59

Capitolo 3, Problema 59

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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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.

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Function Composition

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.

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Evaluating Composed Functions

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.

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Multiplying & Dividing Functions

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