Example Card: Determining the Domain of a Square-Root Function

Before we can graph a square root function, we must know where it even exists! This example shows how to find the function's valid starting point, which is a key idea from this lesson.

Example Problem Determine the domain of the function $y = 5\sqrt{\frac{x}{3} + 2} 9$.

Step by Step 1. The domain is the set of all possible input values for $x$. For a square root function to be a real number, the expression inside the radical (the radicand) cannot be negative. 2. We set the radicand to be greater than or equal to zero. The terms outside the radical, like the $5$ and $ 9$, do not affect the domain. $$ \frac{x}{3} + 2 \ge 0 $$ 3. To solve for $x$, first subtract $2$ from both sides of the inequality. $$ \frac{x}{3} \ge 2 $$ 4. Next, multiply both sides by $3$ to isolate $x$. $$ x \ge 6 $$ 5. The domain is the set of all real numbers greater than or equal to $ 6$. This means the graph will start at $x = 6$ and extend to the right.