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

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:

Evaluate $ f(4) $
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 $

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📰 Solution: The first 5 terms are $ a_1 = 5 $, $ a_2 = 8 $, $ a_3 = 11 $, $ a_4 = 14 $, and $ a_5 = 17 $. Their sum is $ 5 + 8 + 11 + 14 + 17 = 55 $. Dividing 55 by 7 gives a quotient of 7 and a remainder of $ 55 - 7 \times 7 = 55 - 49 = 6 $. 📰 Question: What is the least common multiple of the number of eggs laid by three different snake species, which are 12, 18, and 30? 📰 Solution: Factorize each number: $ 12 = 2^2 \times 3 $, $ 18 = 2 \times 3^2 $, and $ 30 = 2 \times 3 \times 5 $. The LCM is the product of the highest powers of all primes: $ 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180 $. 📰 They Alter Realityno One Knows The Truth Behind Them 📰 They Always Said It Was Just Noisebut The Smashing Machine Whispered Truths You Cant Ignore 📰 They Are Mutant Ninjas Secrets Trapped Beneath The Oceans Darkest Reefs 📰 They Arent Just Borrowingtheyre Building Prosperity With A Financial Credit Union 📰 They Arent Just Charmingtheyre Masterful And Youre About To Learn How 📰 They Arent Just Cooking Chicken The Magical Secret Behind Vivas Flavor 📰 They Arent Just Teaching Skillsthis Sper Escuela Of Heroes Is Crushing Fear And Building Legendary Minds 📰 They Arose From The Shadowsstrangers From Hell Are Here And No One Is Safe 📰 They Ate Stuff You Never Imagined In Special Forces Season 3Heres The Truth 📰 They Ate The Burger Secret Of The Villageand The Town Was Never The Same 📰 They Been Timeguessr For Yearsnow The Shocking Truth Revealed 📰 They Broke The Chains Now Theyve Changed Everything You Thought You Knew About The Let Them Theory 📰 They Call Her Spoiledbut Her Words Take You Straight To Shocking Truths 📰 They Call Her The Cleaning Lady But Her Movie Will Change Everything About You 📰 They Called Her Gooduntil The Truth Went Dark

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.