The problem asks for the domain of $f(g(x))$ where $f(x) = \sqrt{x}$ and $g(x) = x - 2$.

Step 1: Composition of functions

We are asked to find the domain of $f(g(x))$. Since $f(x) = \sqrt{x}$, for $f$ to be defined, the argument of the square root must be non-negative: $g(x) = x - 2 \geq 0$

Step 2: Solving the inequality

To ensure $g(x) = x - 2 \geq 0$, solve for $x$: $x - 2 \geq 0 \implies x \geq 2$

Step 3: Conclusion

Thus, the domain of $f(g(x))$ is $x \geq 2$. The correct answer is:

• c. $x \geq 2$.

Do you have any questions, or would you like a further breakdown?

Here are five related questions for further exploration:

1. What is the domain of $f(x) = \sqrt{x-3}$?
2. How do you find the domain of a composition of two functions?
3. If $g(x) = x^2 - 4$, what would be the domain of $f(g(x))$ if $f(x) = \sqrt{x}$?
4. How does the range of $g(x)$ affect the domain of $f(g(x))$?
5. How would you approach finding the range of $f(g(x))$ in this case?

Tip: When working with compositions involving square roots, always make sure the expression inside the root is non-negative for the function to be valid.