Verified answer

Domain:

The denominator can't be zero

x^2 + x - 2 <> 0

x^2 + 2x - x - 2 <> 0

x(x + 2) - (x + 2) <> 0

(x - 1)(x + 2) <> 0

x - 1 <> 0 and x + 2 <> 0

x <> 1 and x <> -2

x can be anything except 1 and -2

The domain is (-infinity, -2) U (-2, 1) U (1, infinity)

Range:

y = (2x - 4) / (x^2 + x - 2)

y(x^2 + x - 2) = 2x - 4

yx^2 + xy - 2y = 2x - 4

yx^2 + xy - 2y - 2x + 4 = 0

yx^2 + (y-2)x + (4-2y) = 0

Use the Quadratic Formula to solve for x:

x = (-(y-2) +/- sqrt((y-2)^2 - 4(y)(4-2y))) / (2y)

x = (-y + 2 +/- sqrt(y^2 - 4y + 4 - 16y + 8y^2)) / (2y)

x = (-y + 2 +/- sqrt(9y^2 - 20y + 4)) / (2y)

If x is a real number, then the discriminant can't be negative:

9y^2 - 20y + 4 >= 0

9y^2 - 2y - 18y + 4 >= 0

y(9y - 2) - 2(9y - 2) >= 0

(y - 2)(9y - 2) >= 0

y <= 2/9 or y >= 2

The range is (-infinity, 2/9] U [2, infinity)