`y=(x-1)/(x+5)` Graph the function. State the domain and range.

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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.



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 the degree of 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.



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

Next, find the intercepts.




So the y-intercept is  `(0, -1/5)` .






So, the x-intercept is `(1,0)` .

Also, determine the other points of the function. To do so, assign any values to x, except -5. And solve for the y values.

`x=-15, y=(-15-1)/(-15+5) = (-16)/(-10)=8/5`

`x=-11, y=(-11-1)/(-11+5)=(-12)/(-6)=2`

`x=-7, y=(-7-1)/(-7+5)=(-8)/(-2)=4`

`x=-6, y=(-6-1)/(-6+5)=(-7)/(-1)=7`

`x=-3, y=(-3-1)/(-3+5) = (-4)/2=-2`

`x=4, y=(4-1)/(4+5)=3/9`

`x=15, y=(15-1)/(15+5)=14/20=7/10`

Then, plot the points `(-15,8/5)` ,   `(-11,2)` ,   `(-7,4)` ,   `(-6,7)` ,   `(-3,-2)` ,   `(0,-1/5)` ,   `(1,0)` ,   `(4,3/9)` and `(15,7/10)` .

And connect them.

Therefore, the graph of the function is:

Base on the graph, the domain of the function is `(-oo, -5) uu (-5,oo)` . And its range is `(-oo, 1) uu (1,oo)` .

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