Problem 22 - Precalculus Essentials

Question

Find the domain of each function. $$g(x)=\sqrt{7 x-70}$$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = \sqrt{7x - 70}\) is [10, +∞).

Step 1: Set the expression under the square root greater or equal to zero

First, set the expression under the square root greater than or equal to zero. So, write the inequality \(7x - 70 \geq 0\). This inequality must be solved to find the values of \(x\) that satisfy it.

Step 2: Solve the inequality

To solve the inequality, it's possible to treat it as an equation first. So, add 70 to both sides to isolate \(7x\) on one side: \(7x = 70\). Next, to isolate \(x\), divide both sides by 7: \(x = 70/7 = 10\). Now, remember it was an inequality, not an equation. Therefore, the solution is \(x \geq 10\), meaning all the values equal to and greater than 10.

Step 3: State the domain

Finally, since \(x \geq 10\), the domain of the function \(g(x) = \sqrt{7x - 70}\) is all real numbers equal to 10 and greater than 10. In interval notation, the domain can be expressed as [10, +∞).