When Does An Inverse Function Exist?

I have understood that an inverse of a function exists when the relation is one to one. It is clear to understand when the function is a whole number. But how do you apply that rule to an equation? What are the steps involved?

The book directly answered that:

**f(x) = 3x - 4** *the inverse of the function exists*

**f(x) = (2x - 3) / (x -7); x is not equal to 7** *the inverse of the function does not exist*

**f(x) = (x^2) - 1** *the inverse of the function does not exist*

What are the steps in between?