fogoh(x) = f(g(h(x)))

fogoh(x) = f(g(h(x)))

The Correct Answer and Explanation is:

To find fogoh(x) = f(g(h(x))), you need to understand function composition step by step.

Step 1: Understand Function Composition

Function composition means applying one function to the results of another function. The notation fogoh(x) represents applying h(x) first, then plugging that result into g(x), and finally plugging that into f(x).

The expression breaks down like this:
fogoh(x) = f(g(h(x)))

This means:

1. Start with x and apply the function h, so you get h(x)
2. Take that result and apply the function g, giving g(h(x))
3. Finally, apply the function f to the result, which gives f(g(h(x)))

Step 2: Example with Defined Functions

Suppose:
h(x) = x + 1
g(x) = 2x
f(x) = x²

We are asked to find fogoh(x), which means f(g(h(x))).

First, compute h(x):
h(x) = x + 1

Second, compute g(h(x)):
g(h(x)) = g(x + 1) = 2(x + 1) = 2x + 2

Third, compute f(g(h(x))):
f(g(h(x))) = f(2x + 2) = (2x + 2)² = 4x² + 8x + 4

Thus, the final answer is:
fogoh(x) = 4x² + 8x + 4

Step 3: Importance of Function Composition

Function composition allows complex operations by combining simpler functions. In mathematics, this is used in algebra, calculus, and real-world modeling. Each function represents a transformation or a process, and composing them shows how several processes affect a quantity step by step.

The order matters greatly in composition. Applying h, then g, then f is different from changing the order. You must carefully follow the sequence to avoid mistakes.

In conclusion, fogoh(x) = f(g(h(x))) means applying functions stepwise in the given order to compute the final output accurately.