To solve $(h \circ g \circ f)(x)$, also written as $h(g(f(x)))$, we will substitute functions step by step.

Step 1: Find $f(x)$

Given $f(x) = 7 + x$, substitute it into the next function.

Step 2: Find $g(f(x))$

Now, substitute $f(x)$ into $g(x)$, which is $g(x) = x^2 - 2$: $g(f(x)) = g(7 + x) = (7 + x)^2 - 2$ Expanding $(7 + x)^2$: $(7 + x)^2 = 49 + 14x + x^2$ Thus, $g(f(x)) = 49 + 14x + x^2 - 2 = x^2 + 14x + 47$

Step 3: Find $h(g(f(x)))$

Now, substitute $g(f(x))$ into $h(x)$, which is $h(x) = 3x$: $h(g(f(x))) = h(x^2 + 14x + 47) = 3(x^2 + 14x + 47)$ Expanding: $h(g(f(x))) = 3x^2 + 42x + 141$

Thus, $(h \circ g \circ f)(x) = 3x^2 + 42x + 141$.

Do you want more details or have any questions? Here are some related questions:

1. How do we compose functions in reverse order?
2. What happens if we change the order of composition in $(f \circ g \circ h)(x)$?
3. How can we visualize function composition graphically?
4. What is the domain of the composed function $(h \circ g \circ f)(x)$?
5. How does function composition relate to transformations in geometry?

Tip: When expanding binomials like $(7 + x)^2$, remember the pattern $a^2 + 2ab + b^2$.