Question

What is the domain and range of `y=-x^2+4x-1`?

Answer 1

Domain: `x in RR`
Range: `y in (-oo,3]`

Explanation:

This is a polynomial, so the domain (all possible `x` values for which `y` is defined) is all real numbers, or `RR`.

To find the range, we need to find the vertex.

To find the vertex, we need to find the axis of symmetry.

The axis of symmetry is `x = -b/(2a) = -4/(2*(-1)) = 2`

Now, to find the vertex, we plug in `2` for `x` and find `y`.

`y = -(2)^2+4(2)-1`

`y = -4+8-1`

`y = 3`

The vertex is either the maximum or minimum value, depending on whether the parabola faces up or down.

For this parabola, `a = -1`, so the parabola faces down.

Therefore, `y=3` is the maximum value.

So the range is `y in (-oo,3]`