Question: If $ f(x) = 3x - 7 $ and $ g(x) = x^2 + 2x $, what is $ g(f(4)) $? - Treasure Valley Movers

How to Calculate $ g(f(4)) $: A Step-by-Step Guide for Function Composition

Understanding function composition is a fundamental skill in algebra, especially when working with expressions like $ f(x) = 3x - 7 $ and $ g(x) = x^2 + 2x $. One common question that arises is: If $ f(x) = 3x - 7 $ and $ g(x) = x^2 + 2x $, what is $ g(f(4)) $?

This article walks you through the entire process clearly and efficiently, helping you master evaluating composite functions.

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What Is Function Composition?

Function composition involves plugging the output of one function into another. The notation $ g(f(x)) $ means you first apply function $ f $ to $ x $, then take that result and apply function $ g $ to it.

In this case, we need to compute $ g(f(4)) $, which requires two key steps:

1. Evaluate $ f(4) $
   
2. Use that result as the input to $ g(x) $

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Step 1: Evaluate $ f(4) $

Given:
$$
f(x) = 3x - 7
$$

Substitute $ x = 4 $:
$$
f(4) = 3(4) - 7 = 12 - 7 = 5
$$

So, $ f(4) = 5 $

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Step 2: Evaluate $ g(f(4)) = g(5) $

Now use the result from Step 1. Since $ g(x) = x^2 + 2x $, substitute $ x = 5 $:
$$
g(5) = 5^2 + 2(5) = 25 + 10 = 35
$$

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Final Answer

Putting it all together:
$$
g(f(4)) = g(5) = 35
$$

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Why This Matters: Function Composition in Math and Beyond

Function composition is not just a theoretical concept — it appears in chemistry, economics, computer science, and engineering. For example, if $ f(x) $ models a price increase and $ g(x) $ represents tax calculation, then $ g(f(x)) $ combines both steps sequentially.

Mastering these principles helps build problem-solving skills essential in higher mathematics, programming, and real-world modeling.