Yes, the inverse would be a function. Allow me to explain.
A function has an input x and an output y. An inverse function simply reverses the function. So for a 1:1 function, for every time x goes up by one, y also goes up one. So for the inverse of the function, it is the opposite. If I am not mistaken, every time y would go up one, x would go down one, and vice versa. That is the only part I am not 100% sure on, so I'd stick with the answer shown below.
Answer: Yes, the inverse of a 1:1 function would also be a function.