which graph shows a function whose inverse is also a function?

Graph 3

Step-by-step explanation:

A function is a relation between to sets, Domain and Range, such that for every element in the domain there is one and only one image in the range. This definition implies basically that, given the x's in the domain, any value x on the domain must have only one image. Thus, lets analyze the graphs and identify which has a f^(-1)(x) that fulfills this condition.

In the first we can see that, for example, x=-3 has two images one of about -0.3 and other of about 0.5. We notice that because the curve passes trough x=-3 more than one time. So, this is not a function.

In the second. Again, we can see that there are values of x with two images. More precisely, every x < 1 has to images, one positive and one negative. If you draw a vertical line in every x<1 you can see it will cut the curve twice, while a function only does it once.

In the third we do have a function. Pick f^-1 and try to draw a vertical line in every x, you will see that it only cuts f^1 once, so every x has only one image. This is because we are working with linear functions. Linear functions (with slope different from 0) always have inverse functions in its domain.

Finally, the fourth graph does not has a inverse function. Similar to what happened in graph 2, take every x > 1 and see that those x will have 2 images by tracing a vertical line in every x.

So, only graph 3 has an inverse function.