When we say that a function is continuous between x = -3 and x = 0 it means that it exists for all values of x in between -3 and 0. Let's take a look at each choice individually:
A: f(x) = (-x + 1)/(x + 2) Now we don't actually need to know what the graph of this function looks like to see which values it is continuous for, instead we should look at which values of x will make this function undefined - in this case that would be x = -2. The reasoning behind this is that a number divided by 0 would be undefined, so when we search for which value of x would make the denominator of the equation 0, we get: x + 2 = 0 x = -2 Since x = -2 is within the interval [-3, 0] we cannot say the function is continuous over this interval
B: f(x) = -2/(x + 1) Using the same method as above we get: x + 1 = 0 x = -1 x = -1 is again within the interval [-3, 0] and so the function is not continuous within this interval
C: f(x) = 3x/(x - 2) x - 2 = 0 x = 2 x = 2 is outside the interval of [-3, 0] and so the function is continuous within this interval and C is the correct answer.
Just for the sake of it however we can look at D as well: D: f(x) = 1/(2x + 1) 2x + 1 = 0 x = -1/2 -1/2 is within [-3, 0] and so D is not continuous over this interval
