The equation can be of the form y=(ax+b)/(cx+d).
cx+d=0 when x=1 (not x+1), so c+d=0, making d=-c. (y=x+1 is not vertical.)
The horizontal asymptote is found by replacing x with a large value X, so y=(aX+b)/(cX+d)=aX/cX=a/c when X is very large, so a/c=¾, so 4a=3c and c=4a/3.
The y-intercept is found by setting x=0: b/d=2, making d=b/2.
Since d=-c=-4a/3, and d=b/2, b/2=-4a/3, so b=-8a/3.
If a=3, b=-8, c=4 and d=-4, making y=(3x-8)/(4x-4). The graph is shown below (green), and its asymptotes are shown in red.