Question 92797
 
{{{f(x)= (x^2 + x - 12)/((x-3)(x+5))}}}


First simplify it:

factor the numerator. we get

{{{(x+4)(x-3)/((x-3)(x+5))}}},

cancel the common factor x-3

we get {{{(x+4)/(x+5)}}}

for a rational expression, the denominator can not be 0,

so x can not equal to -5

the domain is any real number except -5.

The range: Let's simplify {{{(x+4)/(x+5)}}} further

{{{(x+4)/(x+5)}}} = {{{(x+5-1)/(x+5)}}}
={{{1-1/(x+5)}}}

let's take a look at the fraction {{{1/(x+5)}}}

when x is in the neighborhood of -5, the fraction can be positive infinity and 
negative infinity, and thus {{{1-1/(x+5)}}} can be both positive infinity and
negative infinity.

so the range is from negative infinity to positive infinity.

this conclusion can be verified by the graph of {{{f(x)= (x^2 + x - 12)/((x-3)(x+5))}}} below:

{{{graph(200,200, -5.05,-4.95, -1000, 1000,  (1-1/(x+5)))}}}