To be able to graph the rational function `y =(x+4)/(x-3)` , 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 =(x+4)/(x-3)` , the `D(x) =x-3.`

Then, `D(x) =0 ` will be:

`x-3=0`

`x-3+3=0+3`

`x=3`

The **vertical asymptote** exists at `x=3` .

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 =(x+4)/(x-3)` , the leading terms are `ax^n=x or 1x^1` and `bx^m=x or 1x^1` . The values `n =1` and `m=1` satisfy the **condition: n=m**. Then, **horizontal asymptote** exists at` y=1/1 ` or `y =1` .

To solve for possible y-intercept, we plug-in `x=0` and solve for `y` .

`y =(0+4)/(0-3)`

`y =4/(-3)`

`y = -4/3 or -1.333` (approximated value)

Then, **y-intercept** is located at a point `(0, -1.333)` .

To solve for possible x-intercept, we plug-in `y=0` and solve for `x` .

`0 =(x+4)/(x-3)`

`0*(x-3) =(x+4)/(x-3)*(x-3)`

`0 =x+4`

`0-4=x+4-4`

`-4=x or x=-4`

Then, **x-intercept** is located at a point `(-4,0)` .

Solve for **additional points** as needed to sketch the graph.

When `x=2` , the `y = (2+4)/(2-3)=6/(-1)=-6` . point: `(2,-6)`

When `x=4` , the `y =(4+4)/(4-3) =8/1=8` . point:` (4,8)`

When `x=10` , the `y =(10+4)/(10-3)=14/7=2` . point: `(10,2)`

When `x=-16` , the `y =(-16+4)/(-16-3)=-12/(-19)~~0.632` . point: `(-16,0.632)`

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, 3)uu(3,oo)`

and **range**: `(-oo,1)uu(1,oo).`

The domain of the function is based on the possible values of `x.` The `x=3` excluded due to the vertical asymptote.

The range of the function is based on the possible values of `y` . The `y=1` is excluded due to the horizontal asymptote.

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