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.

Understanding the Context

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

Key Insights

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 $

Continue Reading

Match:
\(x^2\): \(b - a + 1 = -2 \Rightarrow b - 4 + 1 = -2 \Rightarrow b = 1\)
\(x\): \(a - b + A = 1 \Rightarrow 4 - 1 + A = 1 \Rightarrow A = -2\)

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Final Thoughts

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

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

Final Answer

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

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.