To solve $(h \circ g \circ f)(x)$, we need to find the composite function:

$(h \circ g \circ f)(x) = h(g(f(x)))$

Step-by-step solution:

1. Start with $f(x) = 7 + x$: $f(x) = 7 + x$

2. Now substitute $f(x)$ into $g(x) = x^2 - 2$: $g(f(x)) = g(7 + x) = (7 + x)^2 - 2$ Expand $(7 + x)^2$: $(7 + x)^2 = 49 + 14x + x^2$ So: $g(f(x)) = 49 + 14x + x^2 - 2 = x^2 + 14x + 47$

3. Now substitute $g(f(x))$ into $h(x) = 3x$: $h(g(f(x))) = h(x^2 + 14x + 47) = 3(x^2 + 14x + 47)$ Distribute the 3: $h(g(f(x))) = 3x^2 + 42x + 141$

Thus, the composite function $(h \circ g \circ f)(x)$ is:

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

Would you like further details or clarifications?

Here are some related questions:

1. How would you compute $(g \circ h)(x)$?
2. What is the domain of $(h \circ g \circ f)(x)$?
3. Can you compute the inverse of $(h \circ g \circ f)(x)$?
4. How would you graph $(h \circ g \circ f)(x)$?
5. What happens if we change $f(x) = 7x$? How does $(h \circ g \circ f)(x)$ change?

Tip: Always expand expressions carefully, especially when dealing with compositions of functions.