Answer:
It is continuous since 
Step-by-step explanation:
We are given that the function is defined as follows 
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 
. 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)
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.
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.
Thus, 
, so by definition, f is continuous at x=0, hence continuous over the interval [-4,5].
