We are asked to find the composition of functions $f \circ (g \circ h)(x)$, where:

• $f(x) = x + 4$,
• $g(x) = 2 - x$,
• $h(x) = x^2 - x + 1$.

Let's break it down step by step:

Step 1: Find $(g \circ h)(x)$

The function $g \circ h(x)$ means $g(h(x))$, or applying $g$ to $h(x)$.

Since $h(x) = x^2 - x + 1$, we substitute this into $g(x) = 2 - x$: $g(h(x)) = 2 - (x^2 - x + 1) = 2 - x^2 + x - 1 = 1 - x^2 + x.$ Thus, $g \circ h(x) = 1 - x^2 + x$.

Step 2: Find $f \circ (g \circ h)(x)$

Now, we apply $f$ to the result of $g \circ h(x)$. That is, $f(g \circ h(x)) = f(1 - x^2 + x)$.

Substitute $1 - x^2 + x$ into $f(x) = x + 4$: $f(1 - x^2 + x) = (1 - x^2 + x) + 4 = -x^2 + x + 1 + 4 = -x^2 + x + 5.$

Thus, $f \circ (g \circ h)(x) = -x^2 + x + 5$.

Final Answer:

$f \circ (g \circ h)(x) = -x^2 + x + 5.$

Would you like further details or clarifications?

Here are 5 related questions to explore:

1. What is the difference between composing two functions and multiplying them?
2. How do you find the inverse of a composite function?
3. Can you explain the domain and range of $f \circ g(x)$?
4. How does composition affect the order of operations in functions?
5. How would the result change if $f(x) = x^2 + 4$ instead?

Tip: When composing functions, always apply the innermost function first, then move outward.