I have no idea what your textbooks say, so I'll say it in my own way.

Suppose you have a function f which is continuous, one-to-one, and onto on a fixed interval [a,b] and you want to find its inverse. The inverse is defined to be a function g(x) such that g(f(x)) = x.

If we let f(x) = y that means that g(y) = x. Hence f(g(y)) = f(x) = y. so by the above definition f(x) and g(x) are inverse functions of one another.

f(x) sends x -> y

g(y) sends y -> x

the conditions one to one and onto guarantee unique images and pre-images. continuity is for simplicity. not needed, but if a function is discontinous you would express it as a split function and find an inverse for each segment.