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 \(a^2 + 2ab + b^2\).

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

Learn how to solve the function composition (h ∘ g ∘ f)(x) where f(x) = 7 + x, g(x) = x² - 2, and h(x) = 3x. This detailed guide covers polynomial expansion, function composition, and key algebraic concepts. Suitable for students in Grades 9-12.

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