Question 1039733
2x^2 +8x -24/ x^2 +x -12
?


(2x^2 +8x -24)/(x^2 +x -12)


{{{(2x^2+8x-24)/(x^2+x-12)}}}


{{{(2(x^2+4x-12))/((x-3)(x+4))}}}


{{{2((x-2)(x+6))/((x-3)(x+4))}}}


You want to see what happens near the critical values of x to find the set of possible values for the expression (the range).  The expression is undefined for x=3 and for x=-4.  Signs just change at the zeros which are x=2 and x=-6.  No factor shared in both numerator and denominator, so no missing point.  Obviously NOT continuous at x=3 or x=-4.


INTERVALS to check for signs:
(-infin,-6]
([-6,-4]
[-4,2]
[2,3]
[3,infinity)


You can decide how to check how the values tend through whatever arithmetic you may be learning for handling rational functions, but I will just show a graph here for help in checking.


{{{graph(300,300,-8,6,-7,7,(2x^2+8x-24)/(x^2+x-12))}}}




The RANGE for the rational expression is ALL REAL NUMBERS.