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

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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.

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