Problem 20 - Precalculus Essentials

Question

Find the domain of each function. $$g(x)=\frac{1}{\sqrt{x+2}}$$

Step-by-Step Solution

Answer

The domain of the function \(g(x)=\frac{1}{\sqrt{x+2}}\) is \((-2, \infty)\).

Step 1: Identify the denominator of the function and set the conditions

The denominator of the function is \(\sqrt{x+2}\). The square root is only defined for positive numbers and zero. Also, a fraction is not defined when the denominator is zero. So \(x+2\) must be greater than zero. Write the inequality: \(x+2 > 0\).

Step 2: Solve the inequality for x

To solve for \(x\), subtract 2 from both sides: \(x > -2\).

Step 3: Define the domain of the function

The domain of the function \(g(x)=\frac{1}{\sqrt{x+2}}\) is all \(x\) such that \(x > -2\). In interval notation, the domain is \((-2, \infty)\).