How do you show that #f(x)=2x# and #g(x)=x/2# are inverse functions algebraically and graphically?

Algebraically:

Two functions are inverses when making the input of one function the output of the other creates the identity function.

Take any input #x#. Feed it to #f#, and #f# returns #f(x)#.
Now, take that #f(x)# and feed it to #g#. Then #g# returns #g[f(x)]#.
If this final output also happens to be #x#, no matter what, then #f# and #g# are inverses.

In math terms, if #f[g(x)]=g[f(x)] = x#, then #f# and #g# are inverses.

Try taking the output of #f# and feeding it to #g#:

#f(x)=2x#

#=>g(color(blue)(f(x)))=g(color(blue)(2x))=color(blue)(2x)/2=x#

So #g(f(x))=x#. In essence, whatever #f# did to #x#, the function #g# undid. Thus, #g# is the inverse of #f#.

Graphically:

Two functions are inverses when their graphs are mirror images of each other along the (identity) line #y=x#.

Here is a graph of #f(x)=2x# (the one closer to the #y#-axis) and #g(x)=x/2# (the one closer to the #x#-axis):
graph{(y-2x)(y-x/2)=0 [-10, 10, -5, 5]}

They are reflections of each other along the line #y=x#. Here is the same graph with the line #y=x# added:

graph{(y-2x)(y-x/2)(y-x)=0 [-10, 10, -5, 5]}

As you can see, #y=x# is a line of symmetry for #f# and #g#, and so #f# and #g# are inverses.