Answer:
It is continuous since 
Step-by-step explanation:
We are given that the function is defined as follows  and
 and  and we want to check the continuity in the interval [-4,5]. Note that this a piecewise function whose only critical point (that might be a candidate of a discontinuity)  x=0 since at this point is where the function "changes" of definition. Note that 9-x and 9+12x are polynomials that are continous over all
 and we want to check the continuity in the interval [-4,5]. Note that this a piecewise function whose only critical point (that might be a candidate of a discontinuity)  x=0 since at this point is where the function "changes" of definition. Note that 9-x and 9+12x are polynomials that are continous over all  . So F is continous in the intervals [-4,0) and (0,5]. To check if f(x) is continuous at 0, we must check that
. So F is continous in the intervals [-4,0) and (0,5]. To check if f(x) is continuous at 0, we must check that 
 (this is the definition of continuity at x=0)
 (this is the definition of continuity at x=0)
Note that if x=0, then f(x) = 9-x. So, f(0)=9. On the same time, note that 
 . This result is because the function 9-x is continous at x=0, so the left-hand limit is equal to the value of the function at 0.
. This result is because the function 9-x is continous at x=0, so the left-hand limit is equal to the value of the function at 0. 
Note that when x>0, we have that f(x) = 9+12x. In this case, we have that 
 . As before, this result is because the function 9+12x is continous at x=0, so the right-hand limit is equal to the value of the function at 0.
. As before, this result is because the function 9+12x is continous at x=0, so the right-hand limit is equal to the value of the function at 0. 
Thus,  , so by definition, f is continuous at x=0, hence continuous over the interval [-4,5].
, so by definition, f is continuous at x=0, hence continuous over the interval [-4,5].