Given the function \( f(x) = 2x + 3 \) and \( g(x) = x^2 - 1 \), find the value of \( g(f(4)) \).
["# How to Find ( g(f(4)) ) for the Functions ( f(x) = 2x + 3 ) and ( g(x) = x^2 - 1 )", "When working with composite functions, one common task is to evaluate expressions like ( g(f(x)) ). In this article, we’ll walk through the step-by-step process of finding the value of ( g(f(4)) ) using the functions:", "[\nf(x) = 2x + 3 \quad \ ext{and} \quad g(x) = x^2 - 1\n]", "Understanding how to compose functions and evaluate them correctly is essential in algebra, calculus, and applied mathematics.", "## Step 1: Evaluate the Inner Function ( f(4) )", "To compute ( g(f(4)) ), we first evaluate ( f(4) ):", "[\nf(4) = 2(4) + 3 = 8 + 3 = 11\n]", "So, ( f(4) = 11 ).", "## Step 2: Use the Result in the Outer Function ( g(x) )", "Now that we have ( f(4) = 11 ), we substitute this value into the function ( g(x) ):", "[\ng(f(4)) = g(11)\n]", "[\ng(11) = (11)^2 - 1 = 121 - 1 = 120\n]", "## Final Result", "Therefore, the value of ( g(f(4)) ) is:", "[\n\boxed{120}\n]", "---", "TL;DR:\nTo find ( g(f(4)) ):\n1. Calculate ( f(4) = 2(4) + 3 = 11 )\n2. Then substitute into ( g(x) ): ( g(11) = 11^2 - 1 = 120 )\nAnswer: ( g(f(4)) = 120 )", "Understanding this process helps build confidence in working with function composition, a fundamental concept in higher-level math and programming topics such as function pipelines and data transformations."]
