Question

Find the domain of \(y=\frac{3}{\sqrt{x}-2}\)

Short answer

The domain of the function \(y=\frac{3}{\sqrt{x}-2}\) is \(0 \leq x < 4\) and \(x > 4\).

Step-by-step solution

Solve \(\sqrt{x} - 2 \neq 0\)

Firstly, solve the equation \(\sqrt{x} - 2 \neq 0\) to find the values of x that will not result in zero in the denominator. This step involves balancing the equation by adding 2 to both sides and then squaring both sides to eliminate the square root. Solving this gives \(x \neq 4\).

Solve \(\sqrt{x} \geq 0\)

Next, solve \(\sqrt{x} \geq 0\) to find the values of x that will not result in the square root of a negative number. The square root of zero is defined and is zero, so this results in \(x \geq 0\).

Find Intersection of the Solution Sets

The final step is to find the intersection of the solutions found in Step 1 and Step 2. This is because the domain of the function is defined by x values that satisfy both conditions (\(\sqrt{x} - 2 \neq 0\) and \(\sqrt{x} \geq 0)\). The intersection of \(x \geq 0\) and \(x \neq 4\) is \(0 \leq x < 4\) and \(x > 4\). This means the function is defined for all x values greater than or equal to 0 but not equal to 4.