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