analyze the graph of the function f(x)=x^2-x-2 / x+4 a. domain b.verticle asympote x= c. horizontal or oblique y= d. graph

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a. You need to remember that the domain of the function consists of all values of x that makes the function possible. Hence, you need to exclude from domain ofthe function the values that cancel the denominator such that:

`x + 4 != 0 => x != -4`

Hence, evaluating the domain of the function yields the real set `R - {-4}.`

b. Notice that since `x = -4`  is the excluded value, hence, the function has a vertical asymptote at `x = -4` .

c. You need to evaluate the following limits such that:

`lim_(x->+-oo) (x^2 - x - 2)/(x+4) = +-oo`

Since evaluating the limits above yields `+-oo` , the function has no horizontal asymptotes.

You need to evaluate the slant asymptote such that:

`y = mx + n`

`m = lim_(x->+-oo) (x^2 - x - 2)/(x(x+4))`

`m = 1`

`n = lim_(x->+-oo) ((x^2 - x - 2)/(x+4) - x)`

`n = lim_(x->+-oo) (x^2 - x - 2 - x^2 - 4x)/(x+4)`

`n = lim_(x->+-oo) (-5x - 2)/(x+4) = -5`

Hence, evaluating the slant asymptote yields `y = x - 5` .

d. Sketching the graph of the given function yields: