Understanding $ f(g(2)) $: A Step-by-Step Solution Using Functions $ f(x) = x^2 - 4x + 5 $ and $ g(x) = 3x - 1 $

Ever encountered the expression $ f(g(2)) $ and wondered how to evaluate it? Whether you're a student learning function composition or simply curious about working with mathematical functions, solving $ f(g(2)) $ involves two key steps: first, evaluate the inner function $ g(2) $, then use that result as the input for $ f $. In this article, we break down how to compute $ f(g(2)) $ using the specific functions $ f(x) = x^2 - 4x + 5 $ and $ g(x) = 3x - 1 $, making it easier to understand sequence and function operations.

Understanding the Context

What Does $ f(g(2)) $ Mean?

Function composition, denoted $ f(g(x)) $, means applying the function $ g $ first, then applying $ f $ to the result. So $ f(g(2)) $ means plug $ g(2) $ into $ f $. Follow these steps:

  1. Evaluate $ g(2) $
  2. Plug the result into $ f $
  3. Simplify to find $ f(g(2)) $
Question: What is $ f(g(2)) $ if $ f(x) = x^2 - 4x + 5 $ and $ g(x) = 3x - 1 $?

Key Insights

Step 1: Calculate $ g(2) $

Given:
$ g(x) = 3x - 1 $

Substitute $ x = 2 $:
$ g(2) = 3(2) - 1 = 6 - 1 = 5 $

Step 2: Evaluate $ f(5) $

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Final Thoughts

Now that we know $ g(2) = 5 $, compute:
$ f(x) = x^2 - 4x + 5 $
So,
$ f(5) = (5)^2 - 4(5) + 5 = 25 - 20 + 5 = 10 $

Final Result: $ f(g(2)) = 10 $

Thus, $ f(g(2)) = 10 $ is the final value.

Why Understanding Function Composition Matters

Function composition $ f(g(x)) $ is fundamental in algebra, calculus, and applied mathematics. It models real-world scenarios where one process depends on another — such as transforming data through successive operations. Understanding how to decompose $ f(g(x)) $ into manageable steps strengthens your problem-solving skills and prepares you for advanced topics.

Summary

  • $ g(x) = 3x - 1 $
  • $ g(2) = 5 $
  • $ f(x) = x^2 - 4x + 5 $
  • $ f(5) = 10 $
  • Therefore, $ f(g(2)) = 10 $