Given the hyperbola:
`4x^2 - y^2 - 8x -4y +16 = 0`
First we will write into the standard form.
==> Divide by 4.
`==gt x^2 - y^2/4 - 2x - y + 4 = 0`
Now we will complete the squares,
==> `(x^2-2x) - (y^2/4 +y) = -4`
==> `(x-1)^2 - (y+2)^2/4 = -2`
Now we will divide by -2 so the right side is 1
`==gt (y+2)^2/8 - (x-1)^2/2 = 1`
The hyperbola is opening up/down==> Then we have a vertical hyperbola.
1. Domain: we do not have any restrictions on the domain
==> The domain is all real numbers.
2. range: We will find the vertices's.
Vertices's are: `(h, k+-a) = (1, -2+-2sqrt2)`
==> Vertices are:`(1, -2+2sqrt2)` and `(1, -2-2sqrt2) `
==> Then, the range is all real numbers except the interval `(-2-2sqrt2, -2+2sqrt2)` .
==> Range = `R- (-2-2sqrt2, -2+2sqrt2)`