y = (x² + 3x - 1)/(x - 2)

= (x² + 3x - 10 + 9)/(x - 2)

= ((x - 2)(x + 5) + 9)/(x - 2)

= x + 5 + 9/(x - 2)


There is an oblique asymptote y = x + 5 and a vertical asymptote x = 2

y (x - 2) = x^2 + 3x - 1
yx - 2y = x^2 + 3x - 1
0 = x^2 + 3x - yx + 2y - 1
0 = x^2 + (3 - y)x + 2y - 1

For the absence of the graphs,

Discriminant < 0
(3 - y)^2 - 4 (1) (2y - 1) < 0
9 - 6y + y^2 - 8y + 4 < 0
y^2 - 14y + 13 < 0
(y - 1) (y - 13) < 0
1 < y < 13

So y cannot take values between 1 and 13.

The curve has two turning points. There is no part of the curve in the range between the y-coordinates of the two turning points.

dy/dx = 1 - 9/(x - 2)²

When dy/dx = 0,

1 - 9/(x - 2)² = 0

1 = 9/(x - 2)²

(x - 2)² = 9

x - 2 = 3 or - 3

x = 5 or x = -1

y = 5 + 5 + 9/(5 - 2) or y = -1 + 5 + 9/(-1 - 2)

y = 13 or y = 1


y cannot lie between 1 and 13

y ∈ R \ (1,13)