Explain how this function behaves when it approaches the given x values!

This is a piecewise function because it is defined by more than two functions. Basically, we want to take the limit here. Recall that if a function  approaches some value  as  approaches  from both the right and the left, then the limit of exists and equals  . Here we won't calculate the limit, but apply some concepts of it. So:

a.

Move on the x-axis from the left to the right and you realize that as x increases y also increases without bound.

b.

Move on the x-axis from the right to the left and you realize that as x decreases to negative values y approaches zero.

c.

Since the function is continuous here, we can say that

d.

The function is discontinuous here, but exists and equals 0 as the black hole indicates at .

e.

The function is also discontinuous here, but the black hole indicates that this exists at , so

f.

Since the function is continuous here, we can say that