Problem 21 - Precalculus Essentials

Question

Find the domain of each function. $$g(x)=\sqrt{5 x+35}$$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = \sqrt{5x + 35}\) is \(x \geq -7\)

Step 1: Identify the main component of the function

The first step is to recognize that the entire function hangs on the value under the square root sign, which is \(5x + 35\). For the function to have real values, \(5x + 35\) must be either positive or zero.

Step 2: Set an inequality equation

Because we want to find the range of values for 'x' that make the function work, we can set up the inequality \(5x + 35 \geq 0\)

Step 3: Solve the inequality

To solve for 'x', we need to subtract 35 from both sides of the inequality, which gives \(5x \geq -35\). Then, divide both sides by 5 to isolate 'x', resulting in \(x \geq -7\)