Notes for Example 29-4Q: Domain and Range within a Window

You're absolutely right in your understanding up to the point of x^2 le 4! After x^2 le 4, to solve for x, we take the square root of both sides. However, when you take the square root of both sides in an inequality involving a squared variable, you need to consider both the positive and negative roots.

The rule is: if x^2 le a (where a is a positive number), then - sqrt{a} le x le sqrt{a}. This is because squaring a negative number makes it positive, so both x and -x need to satisfy the condition.

In our case, we have x^2 le 4:

1. Take the square root of both sides: sqrt{x^2} le sqrt{4}

2. This simplifies to |x| le 2 (the absolute value of x is less than or equal to 2).

3. The inequality |x| le 2 means that x is any value whose distance from zero is less than or equal to 2. This translates to x being between -2 and 2, inclusive.

So, -2 le x le 2 is how we get the -2 and 2 from x^2 le 4.