We are given two functions:

• $f(x) = \sqrt{x - 1}$
• $g(x) = x^2 - 8$

We are asked to find the domain of the composition $f \circ g$, where $(f \circ g)(x) = f(g(x))$, and its expression.

Step 1: Find the expression for $(f \circ g)(x)$

First, substitute $g(x)$ into $f(x)$: $(f \circ g)(x) = f(g(x)) = \sqrt{g(x) - 1} = \sqrt{(x^2 - 8) - 1} = \sqrt{x^2 - 9}$ Thus, $(f \circ g)(x) = \sqrt{x^2 - 9}$.

Step 2: Find the domain of $(f \circ g)(x)$

For the square root function to be defined, the argument must be non-negative: $x^2 - 9 \geq 0$ Solving this inequality: $x^2 \geq 9 \quad \Rightarrow \quad x \leq -3 \text{ or } x \geq 3$ Thus, the domain of $(f \circ g)(x)$ is $(-\infty, -3] \cup [3, \infty)$.

Step 3: Conclusion

The domain of $f \circ g$ is $(-\infty, -3] \cup [3, \infty)$, and the expression for $(f \circ g)(x)$ is $\sqrt{x^2 - 9}$.

Looking at the options, option d is the correct answer.

Let me know if you need more details or have any questions!

Here are 5 related questions for further exploration:

1. How do we compose two functions?
2. What are the general rules for determining the domain of a composite function?
3. Can we compose functions if their domains do not overlap?
4. What are other examples of composite functions involving square roots?
5. How would the answer change if the function $f(x)$ were different?

Tip: When composing functions, always ensure that the output of the inner function lies within the domain of the outer function!