Solution: First compute $ f(4) = 3(4) - 7 = 12 - 7 = 5 $. Then substitute into $ g(x) $: $ g(5) = 5^2 + 2(5) = 25 + 10 = 35 $. The value of $ g(f(4)) $ is $ \boxed{35} $.
Understanding Function Composition: How to Evaluate $ g(f(4)) $ Step by Step
Function composition is a powerful concept in mathematics that allows us to build complex transformations by combining simpler functions. In this article, we’ll explore a clear example of computing $ g(f(4)) $, using step-by-step calculations to illustrate the process.
Step 1: Compute $ f(4) $
We begin by evaluating the inner function $ f(x) $. Although the exact definition of $ f(x) $ isn’t provided in this example, our scenario assumes a straightforward linear function:
$$ f(x) = 3x - 7 $$
Now substitute $ x = 4 $ into the function:
$$ f(4) = 3(4) - 7 = 12 - 7 = 5 $$
This first evaluation is crucial — it feeds the output of $ f $ directly into the next function.
Step 2: Substitute $ f(4) $ into $ g(x) $
Next, we use the result $ f(4) = 5 $ as the input for the function $ g(x) $. According to the problem, $ g(x) $ is defined as:
$$ g(x) = x^2 + 2x $$
Now substitute $ x = 5 $ into $ g(x) $:
$$ g(5) = (5)^2 + 2(5) = 25 + 10 = 35 $$
Final Result
Putting it all together:
$$ g(f(4)) = g(5) = 35 $$
Thus, the final value is:
$$ oxed{35} $$
Why Function Composition Matters
Mastering function composition is essential in algebra, calculus, computer science, and engineering. It allows us to break down complex problems into manageable parts and build elegant solutions. Understanding how to evaluate expressions like $ g(f(x)) $ step-by-step builds a strong foundation for higher-level math and real-world applications.
If you’re learning or teaching math, practicing such compositional problems reinforces your grasp of how functions interact and depend on one another.
Try It Yourself! Replace $ f(x) $ with other linear functions, like $ f(x) = 2x + 1 $, and compute $ g(f(3)) $. See how changing the inner function affects the final result — a great way to master composition!
