The function given is $f(x) = \frac{x}{\sqrt{a}}$, where $a$ appears to be a constant.

To determine the domain of this function, we need to check for any restrictions on $x$ based on the nature of the function.

1. Square Root Constraint: The square root function is defined as long as the radicand (the expression inside the square root) is non-negative. However, in this case, $\sqrt{a}$ does not depend on $x$, and if $a > 0$, there is no issue regarding the square root for any real $x$. If $a = 0$, the function would be undefined due to division by zero.

2. Division by Zero: The function is undefined if $a = 0$, because this would cause division by zero. But as long as $a > 0$, there is no issue with division by zero.

Thus, the domain of the function $f(x) = \frac{x}{\sqrt{a}}$ would be all real numbers as long as $a > 0$, meaning $(-\infty, \infty)$ is the domain of the function.

Correct Answer:

The correct choice is (d) $(-\infty, \infty)$, as highlighted in the image.

Would you like further clarification on this?

Here are 5 related questions you might find helpful:

1. What would happen to the domain if $a$ were negative?
2. How would the function change if $a$ depended on $x$?
3. Can the range of this function be determined?
4. What if the function was $f(x) = \frac{1}{\sqrt{a}}$, how would the domain change?
5. How do you find the domain for more complex functions?

Tip: Always check for division by zero and square roots of negative numbers when determining domains!