Calculate the rate of change for the following data

Step-by-step explanation:

There are two directions you want to consider, as x goes to infinity and x goes to negative infinity.  Let's look at those one at a time.

If you look at the table and follow the x values given they are already headed toward negative infinity.  They are going up in value basically.  We can't see super far ahead, but we can see from -5 to 3, so we will use that.

Now we look at f(x), and you should notice that it starts going up, increasing from when x=-5 up until -1, when it starts to change.  So there is a turning point here.  Since that's the case we want to look at what happens after -1, if there is another turning point or it just keeps going.  Well here it starts decreasing, as mentioned with the turning point, and t just keeps going to the last value in the table.  As that's the case  we have to imply it keeps decreasing until negative infinity.

so as x goes to infinity f(x) goes to negative infinity.

Now the other way.  To find this out we have to move backwards ont he table.  Similarly, we want to start with that turning point at -1 ad see if there's another.  Well on the table it doesn't change again.  Going in reverse from here the values keep decreasing, so we have to assume they will keep decreasing all the way down to negative infinity.

So similarly to the other way, as x heads toward negative infinity, f(x) does to.  So you pick the option with both of those in the answer.