How do you find the inverse of a function?

So you are asked to find the inverse of a function f(x).
The inverse function is denoted by f ^-1(x).
To help with this we can use the identity f(f ^-1(x))=x.
Now, we need to define y=f ^-1(x).
Example:
f(x)=2x+1
x=f(f ^-1(x))=f(y)=2y+1                    As f(y) is similar to f(x) but with the variable change of x to y
Hence, we need to solve:           
x=2y+1                                                                
x-1=2y                                                  Minus 1 from each side of the equation
½(x-1)=y=f ^-1(x)                                                As we defined f ^-1(x)=y
Therefore, we have found the inverse function: f ^-1(x) = ½(x-1)

We can continue further and find the domain and range of an inverse function using the identities:
Domain f(x) = Range f ^-1(x)
Range f(x) = Domain f ^-1(x)