Question 293389
f(x) = {{{1/(x-2)}}}


g(x) = {{{4/x}}}


Find (fog)(x)


Find (gof)(x)


First we'll do (fog)(x):


(fog)(x) = f(g(x))


g(x) = {{{(4/x)}}}


f(x) = {{{(1/(x-2))}}}


f(g(x)) becomes f({{{(4/x)}}})


You are replacing x with g(x) in f(x) which means you are replacing x with {{{4/x}}} in f(x).


Your equation f(x) becomes f({{{(4/x)}}})


Since f(x) = {{{1/(x-2)}}} and you are replacing x with (4/x), then your equation becomes:


f({{{(4/x)}}}) = {{{1/((4/x)-2)}}}


If you multiply numerator and denominator of this expression by x, then you get:


f({{{(4/x)}}}) = {{{x / (x*((4/x) - 2))}}} which becomes:


f({{{(4/x)}}}) = {{{x / (4-2x)}}}


To test this out, pick a number for x.


Let x = 5.


f(g({{{4/x}}})) = {{{x/(4-2x)}}} = {{{4/(4-8)}}} = {{{4/(-4)}}} = -1


Solve separately to confirm the answer is good.


Solve for g(x) = {{{4/x}}} to get g(4) = {{{4/4}}} = 1


You now have g(x) = 1


Solve for f(g(x)) = f(1) = {{{1/(x-2)}}} = {{{1/(1-2)}}} = {{{1/(-1)}}} = -1


We get a match again, so the function f(g(x)) was translated correctly.


Next we'll do (gof)(x).


That's equivalent to g(f(x)).


f(x) = {{{1/(x-2)}}}


g(x)= {{{4/x}}}


g(f(x)) is therefore equal to {{{4/(1/(x-2))}}}


Multiply numerator and denominator of this equation by (x-2) to get:


g(f(x)) = {{{(4*(x-2))/1}}} = 4x-8


To confirm we'll do g(x) separately, and then do f(g(x)) from the answer, and we'll do f(g(x)) using the combined formula to see if they match.


We'll choose x = 5 again.


g(f(x)) = 4x-8 = 4*5 - 8 = 20 - 8 = 12


Solve for f(x) separately to get f(x) = {{{1/(x-2)}}} = {{{1/(5-2)}}} = {{{1/3}}}


solve for g(f(x)) = g({{{1/3}}}) to get g({{{1/3}}}) = {{{4/(1/3)}}} = {{{4*3}}} = 12.


The answers match.


Those should be your answers:


f(g(x)) = {{{x / (4-2x)}}}


g(f(x)) = 4x-8