Problem

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

Answer

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Answer

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

Steps

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.

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