Solution: First, evaluate $ g(2) = 2^2 - 1 = 4 - 1 = 3 $. Then substitute into $ f(x) $: $ f(3) = 2(3) + 3 = 6 + 3 = 9 $. \boxed{9}

Understanding Nested Function Evaluation: A Step-by-Step Solution

Functions are fundamental building blocks in mathematics and programming, enabling us to model relationships, transform inputs, and solve complex problems step by step. In this article, we explore a clear, structured approach to evaluating composite functions—using a specific example to demonstrate how function composition works and how to arrive at a precise numerical result efficiently.

Step 1: Compute $ g(2) $

We begin with the function $ g(x) $. According to the problem:

$$ g(2) = 2^2 - 1 = 4 - 1 = 3 $$

This evaluation shows how $ g(x) $, defined as squaring the input and subtracting 1, transforms the input step by step.

Step 2: Substitute $ g(2) $ into $ f(x) $

Next, we evaluate $ f(x) $, which is defined in terms of the output of $ g(2) $. The composite function becomes $ f(g(2)) = f(3) $. Given that:

$$ f(x) = 2x + 3 $$

we substitute $ x = 3 $:

$$ f(3) = 2(3) + 3 = 6 + 3 = 9 $$

Final Result

Thus, the full evaluation yields:

$$ oxed{9} $$

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This example illustrates the core principle of nested function evaluation: breaking down complex expressions into sequential computations. By carefully computing intermediate values—like $ g(2) $—and substituting them into subsequent functions, we ensure accuracy and clarity.

Whether in academic math, algorithm design, or software development, mastering such stepwise problem-solving techniques strengthens analytical thinking and prevents errors. Understanding how to evaluate expressions like $ f(g(2)) = 9 $ lays the groundwork for more advanced applications involving function composition.