Question

How do you find the domain and range of `f(x) = sqrt(4 - x²)`?

Answer 1

Domain: `[-2,2]`.
Range: `[0,2]`.

Explanation:

Domain: the root is well defined only if its argument is non-negative.

So, we must solve `4-x^2\ge 0`, which leads to `x^2 \le 4`. This inequality holds for `x \in [-2,2]`.

As for the range, we observe that a root is always positive, so since `f(-2)=f(2)=sqrt(4-4)=0`, we have that `0` is the lowest possible value.

Also, we observe that `f(0)=sqrt(4)=2`, and for every other `x` we will have `f(x)=sqrt(4-x^2)`, which means the square root of something less than `4`, as thus something less than `2`. So, `2` is the maximum of the function.