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!