#f^(-1)(x)# is the inverse function of #f(x)#. All that means is an input for #f(x)# equals an output for #f^(-1)(x)#. For example, the functions #f(x)=-x# and #f^(-1)(x)=x# are inverses. #f(1)=-1# and #f^(-1)(-1)=1#. An inverse function generates the input for the original function.
You can find inverses in 4 simple steps.
Step 1: Change to #x# and #y# Notation
All this means is replace #f(x)# with #y#:
#y=(x+1)/2#
Step 2: Swap #x# and #y#
#x# becomes #y# and #y# becomes #x#:
#x=(y+1)/2#
Step 3: Solve for #y#
We have #x=(y+1)/2#. Solving for #y# is just a matter of algebra:
#x=(y+1)/2#
#2x=y+1#
#2x-1=y#
Step 4: Replace #y# with #f^(-1)(x)#
We change back to #f(x)# notation in this step, adding a #-1# to say it's an inverse:
#f^(-1)(x)=2x-1#
