Find the Domain of the Square Root Function: Simplifying √(4x²-4)

Look at the following function:

$\frac{\sqrt{4x^2-4}}{10}$

What is the domain of the function?

To determine the domain of the function $\frac{\sqrt{4x^2-4}}{10}$, we need to ensure that the expression inside the square root is non-negative. This ensures the function is defined for those values of $x$.

First, set the expression inside the square root to be non-negative:

• $4x^2 - 4 \geq 0$

Next, solve this inequality:

• Factor the expression: $4(x^2 - 1) \geq 0$, which simplifies to $x^2 - 1 \geq 0$.
• Further factorization gives: $(x - 1)(x + 1) \geq 0$.

Now, determine the intervals where this product is non-negative:

• The critical points are $x = 1$ and $x = -1$.
• Test intervals determined by these critical points:
  • Interval $x < -1$: Choose $x = -2$, $(x - 1)(x + 1) = (-3)(-1) = 3 \geq 0$.
  • Interval $-1 \le x \le 1$: Choose $x = 0$, $(x - 1)(x + 1) = (-1)(1) = -1$ which is not $\geq 0$.
  • Interval $x > 1$: Choose $x = 2$, $(x - 1)(x + 1) = (1)(3) = 3 \geq 0$.

Therefore, the solution to the inequality $(x - 1)(x + 1) \geq 0$ is $x \le -1$ or $x \ge 1$.

Thus, the domain of the function is $x \ge 1$ or $x \le -1$.

In the context of the given choices, the solution corresponds to choice 4: $x \ge 1, x \le -1$.

The domain of the function is $x \ge 1$ or $x \le -1$.