Question

Find the asymptotes of the function. Select all that apply.

Answer 1

An asymptote is a vertical horizontal or oblique line to which the graph of a function progressively approaches without ever touching it.

To answer this question we observe the graph. All the values of x and y must be identified for which the graph of the function tends to infinity.

It is observed that these values are:

x = -1

x = 3

y = 0

The first two corresponds to the equations of a vertical line. The third corresponds to horizontal line, the axis of x. It can be seen that although the graph of the function is very close to these values, it never "touches" them

Answer 2

To find asymptotes of the function we look at the graph where it get close to x  and y

We have vertical at x and horizontal asymptotes at y

In the left part of the graph , the graph gets close to y=0 but it does not crosses y=0

So the horizontal asymptote at y=0

Also the graph goes on increasing till x=-1, the graph gets close to -1 but it does not cross x=-1

So the vertical asymptote at x=-1

In the middle part of the graph goes close to x=-1  and x=3

So the vertical asymptote at x=-1 and x=3

In the last part of the graph , the graph is close to x=3  and it does not cross x=3.

also the graph does not cross y=0

So the horizontal asymptote at y=0, the vertical asymptote at x=3

After combining all the asymptotes

x=-1, y=0, x=3