Determine the domain of the function: $\frac{\sqrt{x}}{\sqrt{x}+5}$

Step 1 ：The domain of a function is the set of all possible input values (often the 'x' variable), which produce a valid output from a particular function.

Step 2 ：In this case, we have the function \(\frac{\sqrt{x}}{\sqrt{x}+5}\).

Step 3 ：The domain of this function is determined by two conditions:

Step 4 ：1. The value under the square root must be non-negative (since the square root of a negative number is not a real number). This gives us the condition \(x \geq 0\).

Step 5 ：2. The denominator of a fraction cannot be zero (since division by zero is undefined). This gives us the condition \(\sqrt{x}+5 \neq 0\).

Step 6 ：Let's solve these conditions:

Step 7 ：For \(x \geq 0\), all non-negative real numbers are valid.

Step 8 ：For \(\sqrt{x}+5 \neq 0\), we can solve for x: Subtract 5 from both sides: \(\sqrt{x} \neq -5\). Square both sides: \(x \neq 25\).

Step 9 ：So, the domain of the function is all non-negative real numbers except 25. In interval notation, this is \([0, 25) \cup (25, \infty)\).

Step 10 ：\(\boxed{[0, 25) \cup (25, \infty)}\) is the final answer.