The graph looks to have infinite range, but multiple vertical asymptotes. That means we can limit our choices to tangent and cotangent graphs.

Furthermore, we observe that the graph starts at the bottom and increases from left to right, consistent with tangent graphs. So we narrow our focus to the choices involving tangents.

To decide between the remaining two graphs, observe that y-intercept (where x=0) of our graph is (0,1).

Now   evaluated at  is , which means that we need a vertical shift of  unit.

Hence the best choice is:

To derive the graph of  , recall that .  The graph of   is

and the graph of is

Vertical asymptotes will occur in the graph of   whenever .  This is because the denominator of the tangent function will be equal to zero whenever the cosine function is equal to zero and then the entire function will be undefined at those points.  Wherever cosine crosses the x-axis a vertical asymptote will occur.  If we overlay the sine and cosine graphs we see the following:

So our tangent graph will follow the same form as the sine and cosine graphs when they are increasing, but will have vertical asymptotes wherever cosine crosses the x-axis.

And we are left with our graph of 

We will begin by considering the general graph of  and apply transformations step by step to produce a graph of .  The graph of  is

The general equal of a tangent transformation equation is .  A is the amplitude of the graph of tangent.  Here,  so we do not need to apply a transformation here.  Next, we will consider the period.  The period of the tangent function is equal to .  So the period of our graph would be

Period = 

Period = 

So the period is shortened from  to .

Now, we will consider .   is the phase shift of our graph.  So we will shift our graph  units to the left.  This does not change our graph visually due to the period now being .  Lastly, we will consider .   is the vertical shift of our graph, and so we must shift our graph 1 unit up.

And we are left with the graph of . 

To derive the graph of  recall that .  So the tangent and cotangent graphs are reciprocals of one another.  We will consider the tangent graph since it is one we are more familiar with:

Now we will simply invert the tangent graph to get the cotangent graph

And we are left with our cotangent graph

First, we will consider the graph of  and apply transformations step-by-step.  The graph of  is

The general form of a cotangent transformation function is .  For our function ,  and so we need to increase the amplitude 4 units.

 here so we do not need to make any changes to the period of this graph.  , giving us a negative phase shift of  units.

This leaves us with our graph of .

This is because  and   causing the tangent function to be undefined at these points and forming a vertical asymptote.  This is also true for the cotangent function because  so wherever  is zero or undefined, cotangent will be as well.

 Consider the graph of the function .  It passes the vertical line test, that is if a vertical line is drawn anywhere on the graph it only passes through a single point of the function. This means that  is a function.

Now, for its inverse to also be a function it must pass the horizontal line test. This means that if a horizontal line is drawn anywhere on the graph it will only pass through one point.

This is not true, and we can also see that if we graph the inverse of  () that this does not pass the vertical line test and therefore is not a function.  If you wish to graph the inverse of , then you must restrict the domain so that your graph will pass the vertical line test.

Note that the inverse of  is not , that is the reciprocal.  The inverse of  is  also written as .  The graph of  with  is as follows.

And so the inverse of this graph must be the following with  and 

To find an inverse function you swap the and values.  Take  for example, to find the inverse we use the following method.

 (swap the  and  values)

(solving for )

 If we are looking for the graph of  with , that means this is the inverse of   with .  The graph of  with  is

Switching the  and   values to graph the inverse we get the graph