Domain of y = √(x² - 4) is:

Understand the Problem

The question is asking for the domain of the function y = √(x² - 4). To find the domain, we need to determine the values of x for which the function is defined, specifically where the expression under the square root is non-negative.

Answer

The domain of $y = \sqrt{x^2 - 4}$ is $(-\infty, -2] \cup [2, \infty)$.

Steps to Solve

Set the expression under the square root greater than or equal to zero

We need to find when the expression $x^2 - 4 \geq 0$ for the function to be defined.

Solve the inequality

To solve the inequality, factor the expression:

$$ x^2 - 4 = (x - 2)(x + 2) \geq 0 $$

Now we can find the critical points by setting the factors equal to zero:

$$ x - 2 = 0 \Rightarrow x = 2 $$ $$ x + 2 = 0 \Rightarrow x = -2 $$

Test intervals around the critical points

The critical points divide the number line into intervals. We will test:

Interval 1: $(-\infty, -2)$
Interval 2: $(-2, 2)$
Interval 3: $(2, \infty)$

Choose test points from each interval:

For $(-\infty, -2)$, test $x = -3$: $(-3)^2 - 4 = 9 - 4 = 5 \geq 0$ (valid)
For $(-2, 2)$, test $x = 0$: $0^2 - 4 = -4 < 0$ (not valid)
For $(2, \infty)$, test $x = 3$: $3^2 - 4 = 9 - 4 = 5 \geq 0$ (valid)

Determine the domain

The values that make the original function defined are in the intervals $(-\infty, -2]$ and $[2, \infty)$. Thus, the domain is:

$$ (-\infty, -2] \cup [2, \infty) $$

The domain of $y = \sqrt{x^2 - 4}$ is $(-\infty, -2] \cup [2, \infty)$.

More Information

This result shows the values of $x$ for which the function is defined. The square root function is only defined for non-negative inputs, hence the critical points and interval testing.

Tips

Forgetting to include the endpoints where the square root is zero can lead to a wrong domain.
Misinterpreting the intervals during the testing of values can mislead to incorrect intervals being included.

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