`y=-5/x-7` Graph the function. State the domain and range.

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The given function `y = 5/x-7` is the same as:

`y = 5/x-(7x)/x`

`y = (5-7x)/x or y =(-7x+5)/x.`

To be able to graph the rational function `y =(-7x+5)/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 =(-7x+5)/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 =(-7x+5)/x` , the leading terms are  ` ax^n=-7x or -7x^1` and `bx^m=x or x^1` . The values `n =1` and `m=1` satisfy the condition: n=m. Then, horizontal asymptote  exists at `y=(-7)/1 or y =-7` .

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

`y =(-7*0+5)/0`

`y = 5/0 `  

y = undefined

Thus, there is no y-intercept.

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

`0 =(-7x+5)/x`

`0*x =(-7x+5)/x*x `

`0 =-7x+5`

`0-5=-7x+5-5 `



`x=5/7 or 0.714`  (approximated value)

Then, x-intercept is located at a point `(0.714,0)` .

Solve for additional points as needed to sketch the graph.

When `x=1` , the `y = (-7*1+5)/1=(-2)/1=-2` . point: `(1,-2)`

When `x=5` , the `y =(-7*5+5)/5=(-30)/5=-6` . point: `(5,-6)`

When `x=-1` , the `y =(-7*(-1)+5)/(-1) =12/(-1)=-12` . point: `(-1,-12)`

When `x=-5` , the ` y =(-7*(-5)+5)/(-5) =40/(-5)=-8` . point: `(-5,-8)`

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,-7)uu(-7,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=-7` is excluded due to the horizontal asymptote. 

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