y=4/x+3 Graph the function. State the domain and range.

The given function y = 4/x+3 is the same as:
y = 4/x+(3x)/x
y = (4+3x)/x  or  y =(3x+4)/x .
To be able to graph the rational function y =(3x+4)/x , we solve for possible asymptotes.
Vertical asymptote exists at x=a that will satisfy D(x)=0 on a rational function f(x)=N(x)/D(x) . To solve for the vertical asymptote, we equate the expression at denominator side to 0 and solve for x.
In y =(3x+4)/x, the D(x) =x .
Then, D(x) =0  will be x=0.
The vertical asymptote exists at x=0.
To determine the horizontal asymptote for a given function: f(x) = (ax^n+...)/(bx^m+...) , we follow the conditions:
when n lt m     horizontal asymptote: y=0
        n=m    horizontal asymptote:   y =a/b
        ngtm      horizontal asymptote: NONE
In y =(3x+4)/x , the leading terms are ax^n=3x or 3x^1 and bx^m=x or x^1. Thus, n =1 and m=1 satisfy the condition: n=m . Then, horizontal asymtote  exists at y=3/1  or y =3 .
To solve for possible y-intercept, we plug-in x=0 and solve for y .
y =(3*0+4)/0
y = 4/0
y = undefined
Thus, there is no y-intercept
To solve for possible x-intercept, we plug-in y=0 and solve for x .
0 = (3x+4)/x
0*x = (3x+4)/x*x
0 =3x+4
0-4=3x+4-4
-4 =3x
(-4)/3=(3x)/3
x= -4/3 or -1.333 (approximated value)
Then, x-intercept is located at a point (-1.333,0) .
Solve for additional points as needed to sketch the graph.
When x=2 , then y =(3*2+4)/2 =10/2=5. point: (2,5)
When x=4 , then y =(3*4+4)/4 =16/4=4 . point: (4,4)
When x=-2 , then y =(3*(-2)+4)/(-2) =(-2)/(-2)=1 . point: (-2,1)
When x=-4 , then y =(3*(-4)+4)/(-4) =-8/(-4)=2. point: (-4,2)
 Applying the listed properties of the function, we plot the graph as:
You may check the attached file to verify the plot of asymptotes and points.
As shown on the graph, the domain: (-oo, 0)uu(0,oo)
and Range:  (-oo,3)uu(3,oo)
The domain of the function is based on the possible values of x . The x=0 excluded due to the vertical asymptote
The range of the function is based on the possible values of y. The y=3 is excluded due to the horizontal asymptote.