Question

Question #78544

Answer 1

Domain: `x in RR`
Range: `f(x) in (-oo,+4]`

Explanation:

`f(x)=4-x^2` is defined for all Real values of `x` i.e. for `x in RR`;
actually it is defined for all Complex values of `x` too, but I will assume we are only interested in Reals.

`x^2>=0` for all Real values of `x`
therefore
`-x^2 <= 0` for all Real values of `x`
and
`f(x)=4-x^2 <= 4` for all Real values of `x`

(you might note that as the magnitude of `x` becomes very large there is no lower limit for `f(x)`.

Answer 2

The domain is `(-oo, oo)`
The range is `(-oo, 4]`

Explanation:

`f(x)=4-x^2` => re-write as:
`y=-x^2+4` => equation of an inverted parabola with vertex at:
`(0, 4)` which is the maximum.
The domain is all the real numbers:
`(-oo, oo)`
The range is limited by the y-coordinate of the vertex:
`y<=4`, in interval form:
`(-oo, 4]`