Question

Find the domain and range of the function. (Enter your answers using interval notation.) h(x) = √4 − x^2

Answer 1

The range and domain of this function is [-2, 2]

Explanation:

For the domain it must be fulfilled that
`(4-x^2)\geq 0`. Then,
`(2 + x) (2-x)\geq 0`. This expression is true for the values of
`x` that make the factors simultaneously non-negative or non-positive.

Case non-negative factors

`2 + x \geq 0` implies that
`x \geq -2`, that is
`[-2, +\infty]`

`2-x \geq 0` implies that
`x \leq 2`, that is,
`(-\infty, 2]`. The intersection of the previous sets is [-2, 2].

Case non positive factors

`2 + x \leq 0` implies that
`x \leq -2`, that is
`(-\infty, -2]`

`2-x \leq 0` implies that
`2 \leq x`, that is,
`[2,+\infty)`. The intersection of the previous sets is empty.

Then the domain of the function is [-2, 2]

The range of this function is the domain of its inverse. The inverse of the function is itself, so the range is [-2, 2]