Determine the Domain of the Radical Function: √(2.5x²-5)

Look at the following function:

$\frac{\sqrt{2.5x^2-5}}{5}$

What is the domain of the function?

To solve this problem, first, we determine the condition under the square root function by solving:

$2.5x^2 - 5 \geq 0$.

This inequality ensures that the expression inside the square root is non-negative, a requirement for the square root function to be defined over real numbers.

• Step 1: Simplify the inequality:
• First, add 5 to both sides to isolate the term involving $x$: $2.5x^2 \geq 5$
• Step 2: Solve for $x^2$ by dividing both sides by 2.5: $x^2 \geq \frac{5}{2.5}$
• Step 3: Simplify the fraction: $x^2 \geq 2$
• Step 4: Solve for $x$ by taking the square root of both sides, considering positive and negative solutions: $x \geq \sqrt{2} \quad \text{or} \quad x \leq -\sqrt{2}$

These conditions define the interval for which the original function is defined, corresponding to the original prompt requirement of a non-negative under-the-root value.

Thus, the domain of the function $\frac{\sqrt{2.5x^2-5}}{5}$ is $x \ge \sqrt{2}$ or $x \le -\sqrt{2}$.

The correct choice among the provided options is:

$x \ge \sqrt{2}, x \le -\sqrt{2}$