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We are asked to find the composition \( g \circ h \) of the functions \( g(x) = x^2 - 5 \) and \( h(x) = \sqrt{x - 2} \), and to determine its domain using interval notation.

### Step 1: Find the composition \( g \circ h \)
The composition of functions \( g \circ h \) means we substitute \( h(x) \) into \( g(x) \).

\[
g(h(x)) = g\left(\sqrt{x - 2}\right)
\]

Now substitute \( \sqrt{x - 2} \) into \( g(x) \), which is defined as \( g(x) = x^2 - 5 \):

\[
g(h(x)) = \left( \sqrt{x - 2} \right)^2 - 5
\]

Simplify:

\[
g(h(x)) = (x - 2) - 5 = x - 7
\]

Thus, the composition function is:

\[
g(h(x)) = x - 7
\]

### Step 2: Determine the domain of \( g \circ h \)
The domain of the composition \( g \circ h \) is restricted by the domain of \( h(x) = \sqrt{x - 2} \), because \( g(x) \) is defined for all real numbers, but \( h(x) \) requires that the expression under the square root must be non-negative.

For \( h(x) \) to be defined, we must have:

\[
x - 2 \geq 0
\]

This simplifies to:

\[
x \geq 2
\]

Thus, the domain of \( g \circ h \) is \( [2, \infty) \).

### Final Answer:
- The composition function is \( g(h(x)) = x - 7 \).
- The domain of the composition \( g \circ h \) is \( [2, \infty) \) in interval notation.

Would you like further details or explanations?

Here are some related questions:
1. What would the composition \( h \circ g \) look like for the same functions?
2. How does the square root function restrict the domain of any composition?
3. Can a composition of functions ever have a domain larger than either of the individual functions?
4. How would the domain change if \( g(x) \) or \( h(x) \) had additional constraints?
5. What are some examples where the range of a composition function is restricted?

**Tip:** Always check the domain of both functions before performing composition to ensure the result is defined!

For the real-valued functions g(x) = x^2 - 5 and h(x) = √(x - 2), find the composition g ∘ h and specify its domain using interval notation.

Learn how to find the composition of functions g(x) = x^2 - 5 and h(x) = √(x - 2) and determine its domain using interval notation. This problem covers key concepts in function composition and domain restrictions, suitable for students in Grades 10-12.

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