Problem 19 - Precalculus Essentials

Question

Find the domain of each function. $$g(x)=\frac{1}{\sqrt{x-3}}$$

Step-by-Step Solution

Answer

The domain of the function \(g(x)=\frac{1}{\sqrt{x-3}}\) is \(x > 3\).

Step 1: Understanding the function

The function provided is \(g(x)=\frac{1}{\sqrt{x-3}}\). For this function to be defined: 1.The denominator must not be equal to zero. 2.The value inside the square root should be greater than or equal to zero.

Step 2: Identifying the denominator and the root

In this case, the denominator of the function is \(\sqrt{x-3}\). Hence, this can't be equal to zero. So, \(x - 3 \neq 0\) or \(x \neq 3\). Additionally, the value inside the root should be greater than zero. So, \(x-3 > 0\) or \(x > 3\).

Step 3: Conclusion: Identifying the domain

The domain of the function is \(x >3\). That is, all the values of x greater than 3.