I know the first step is to change it to and then you would switch the x and y variables but when switching the x and y, do you just switch the variables in this case or when switching the variables do you include the of the x?
so would it be
or
sorry, this is posted in the wrong section.
Without the domain, where the function is defined, we cannot really tell if the function has an inverse.
If , the the function has no inverse since the function is not one-one.
To make it a bijective function, you have to first restrict the domain.
I will give an example. Lets say ...
Now its quite evident that f(x) is bijective, so the inverse function exists, call it g(x).
By definition, to find the inverse function g(x), we need
So let , then
Now you will see why we need the bijection condition. As you can see the y that we have got is not a meaningful function. One of the basic properties of a function is that for 1 value of x there must be only one value of f(x).The second property is that it should be defined everywhere in the domain. But here we two values, and .
This is where the domain helps. Assuming , and using the fact that the inverse function can only be in , means is the right choice.
Secondly for to be defined everywhere in the domain, the following should happen:
So we see that is the inverse function.