Problem 20 - College Algebra 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 \( x > -2 \) or in interval notation, \( (-2, +∞) \)

Step 1: Identify the Restrictions

Start by identifying any restrictions on the function. For the function \( g(x)=\frac{1}{\sqrt{x+2}} \), it's clear that \( x+2 \) must be greater than zero (to avoid division by zero) and not a negative number (since the square root of a negative number is not a real number). So the inequality to be solved is \( x + 2 > 0 \).

Step 2: Solve the Inequality

Now, the inequality \( x + 2 > 0 \) is quite simple to solve. One subtracts 2 from both sides to isolate \( x \), resulting in \( x > -2\).

Step 3: Define the Domain

The solution to the inequality defines the domain of the function. So the domain of the function \( g(x)=\frac{1}{\sqrt{x+2}} \) is \( x > -2 \). In interval notation, this would be written as \( (-2, +∞) \).