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.

f(x)=7+x, g(x)=x²-2, h(x)=3x, (hogof)(x)

Learn how to solve the function composition (h ∘ g ∘ f)(x) with step-by-step guidance. This problem involves composing linear and quadratic functions, expanding expressions, and simplifying. Suitable for students in Grades 9-12.

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