https://file.aiquickdraw.com/mathbot/file/1726527089610s1ff95fo.jpg

The functions given are:

\[
f(u) = u + 2
\]
\[
h(u) = u^2 - 3
\]

The problem asks to find \((h \circ h)(u)\), which is the composition of \(h\) with itself, meaning \(h(h(u))\).

### Step-by-step solution:

1. Start by computing \(h(u)\):
   \[
   h(u) = u^2 - 3
   \]

2. Now, substitute \(h(u)\) into \(h\) itself:
   \[
   h(h(u)) = h(u^2 - 3)
   \]

3. Use the expression of \(h(x) = x^2 - 3\), so we substitute \(u^2 - 3\) into this formula:
   \[
   h(u^2 - 3) = (u^2 - 3)^2 - 3
   \]

4. Expand \((u^2 - 3)^2\):
   \[
   (u^2 - 3)^2 = u^4 - 6u^2 + 9
   \]

5. Now substitute this back:
   \[
   h(h(u)) = (u^4 - 6u^2 + 9) - 3
   \]

6. Simplify:
   \[
   h(h(u)) = u^4 - 6u^2 + 6
   \]

Thus, \((h \circ h)(u) = u^4 - 6u^2 + 6\).

Would you like any further details or have questions? Here are some related questions for deeper understanding:

1. How would you compute \((f \circ h)(u)\)?
2. What happens if you reverse the composition to find \((h \circ f)(u)\)?
3. Can you explore the domain and range of \(h(u)\) and \(f(u)\)?
4. What is the behavior of \(h(h(u))\) for large values of \(u\)?
5. How can you find the inverse of \(f(u)\) or \(h(u)\) if they exist?

**Tip:** When dealing with function composition, always substitute the inner function into the outer function carefully, keeping track of all terms!

(h ∘ h)(u) = ? where f(u) = u + 2 and h(u) = u^2 - 3

This problem involves composing functions where h(u) = u^2 - 3 and calculating (h ∘ h)(u). The solution explores polynomial functions and function composition, expanding expressions like (u^2 - 3)^2. Suitable for Grades 10-12 students.

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