`y=(x+6)/(4x-8)` Graph the function. State the domain and range.

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`y=(x+6)/(4x-8)`

First, determine the vertical asymptote of the rational function. Take note that vertical asymptote refers to the values of x that make the function undefined. Since it is undefined when the denominator is zero, to find the VA, set the denominator equal to zero.

`4x-8=0`

`4x=8`

`x=2`

Graph this vertical asymptote on the grid. Its graph should be a dashed line. (See attachment.)

Next, determine the horizontal or slant asymptote. To do so, compare degree of the numerator and denominator.

degree of numerator = 1

degree of the denominator = 1

Since they have the same degree, the asymptote is horizontal. To get the equation of HA, divide the leading coefficient of numerator by the leading coefficient of the denominator.

`y=1/4`

Graph this horizontal asymptote on the grid. Its graph should be a dashed line. (See attachment.)

Next, find the intercepts.

y-intercept:

`y=(0+6)/(4*0-8)=6/(-8)=-3/4`

So the y-intercept is `(0,-3/4)` .

x-intercept:

`0=(x+6)/(4x-8)`

`(4x-8)*0=(x+6)/(4x-8)*(4x-8)`

`0=x+6`

`-6=x`

So, the function has an x-intercept `(-6,0)` .

Also, determine the other points of the function. To do so, assign any values to x, except 1. And solve for the y values.

`x=-10, y=(-10+6)/(4(-10)-8)=(-4)/(-48)=1/12`

`x=-3, y=(-3+6)/(4(-3)-8)=3/(-20)=-3/20`

`x=1, y=(1+6)/(4(1)-8)=7/(-4)=-7/4`

`x=3, y=(3+6)/(4(3)-8)=9/(4)=9/4`

`x=5, y=(5+6)/(4(5)-8) = 11/12=11/12`

`x=6, y=(6+6)/(4(6)-8)=12/16=3/4`

`x=10, y=(10+6)/(4(10)-8)=16/32=1/2`

Then, plot the points `(-10,1/12)` ,   `(-6,0)` ,   `(-3,-3/20)` ,    `(0,-3/4)` ,   `(1,-7/4)` ,   `(3,9/4)` ,   `(5,11/12)` ,   `(6,3/4)` , and `(10,1/2)` .

And connect them.

Therefore, the graph of the function is:

Base on the graph, the domain of the function is `(-oo, 2) uu(2,oo)` . And its range is `(-oo, 1/4) uu (1/4, oo)` .

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