Exploring the Domain of the Function: √(4x² - 8)/5

To determine the domain of the function $\frac{\sqrt{4x^2 - 8}}{5}$, we must ensure the expression under the square root is non-negative. This condition will make the function well-defined over the real numbers.

Steps to solve for the domain:

• Step 1: Set the expression inside the square root greater or equal to zero: $4x^2 - 8 \geq 0$.
• Step 2: Solve the inequality.

To solve the inequality $4x^2 - 8 \geq 0$:

Step 3: Add 8 to both sides:

$4x^2 \geq 8$

Step 4: Divide each term by 4 to simplify:

$x^2 \geq 2$

Step 5: Solve for $x$. When an inequality involves a square, interpret it as involving two cases. Thus, $x \geq \sqrt{2}$ OR $x \leq -\sqrt{2}$.

This inequality describes the values of $x$ for which the function is defined. These constitute the domain of the function. Therefore, the domain is $x \geq \sqrt{2}$ or $x \leq -\sqrt{2}$.

The correct answer choice is:

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