Question

What is the domain of the function f(x)=square root of 3x-21??

Answer 1

The domain of a function is all the x values that it can take.
For example, a horizontal line takes infinite x values (it stretches to negative infinity to the left, and infinity to the right, so the domain is -infinity < x < infinity.

For functions that involve square roots or fractions, you need to remember some rules. Remember that:
- you can't have 0 in the denominator
- you can't have a negative number in a square root

So, considering we have 3x - 21 under the square root, that can't be negative... or you can say that 3x-21 is greater than or equal to 0.
After you solve that inequality, you end up with your domain!

Answer 2

As a slightly different example, say we have the square root of x + 1. That tells us that
\[x + 1 \ge 0 \] which means that \[x \ge -1\] after subtracting 1 from both sides of the inequality. And that would be your domain.
So, our domain is a sort of restriction on the x values (while similarly, the range is the restriction on the y values).

These questions are nice because you can even test if you got the right answer by plugging in some numbers. In my example, I can try putting (-2) + 1 under the square root. That would give me the square root of negative one, which isn't real (try it on your calculator, you'd get some sort of math error). And that makes sense too, because we need our x values to be greater than or equal to -1, while -2 is less than -1.