Problem 60 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

Domain of \(y=\frac{3}{\sqrt{x}-2}\) is \(x\geq0, x\neq4\)

Step 1: Identify Restrictions on the Square Root

Identify all values of \(x\) that would make the value under the square root negative, and exclude these from our potential domain. In this case, \(x\) must be greater than or equal to 0, because \(\sqrt{x}\) is undefined for negative \(x\). So our preliminary domain is \(x\geq0\)

Step 2: Identify Restrictions on the Denominator

Identify all values of \(x\) that would make the denominator zero. This is done by solving \(\sqrt{x}-2=0\), which gives \(x=4\). This means we must exclude 4 from our potential domain, because division by zero is undefined.

Step 3: Combine the Restrictions

We need to find the values of \(x\) that satisfy both restrictions. As per Step 1, \(x\) must be greater than or equal to 0. But according to Step 2, \(x\) cannot be 4. So our final domain is \(x\geq0, x\neq4\).