if f(x)=4/x-1 and g(x)=2x, then the solution set to the equation f(g(x))=g(f(x)) is?

f(x) = 4/(x-1)

g(x) = 2x

f[g(x)] = f(2x) = 4/(2x – 1)

g[f(x)] = g(4/x-1) = 2(4/x-1) = 8/(x-1)

4/(2x-1) = 8/(x-1)

=> 1/(2x-1) = 2/(x-1)

x – 1 = 4x – 2

3x = 1

x = 1/3

f(g(x)) = 4/(2x) – 1 = 2/x – 1
g(f(x) = 2(4/x – 1) = 8/x – 2
so you are asking
2/x – 1 = 8/x – 2
2/x – 8/x = -1
-6/x = -1
x = 6

unless your problem is
f(x) = 4/(x-1) and g(x) = 2x
then
f(g(x)) = 4/(2x – 1)
g(f(x)) = 2(4/(x-1) = 8/(x-1)
so the problem would then be
4/(2x-1) = 8/(x-1)
cross multiply
4(x-1) = 8(2x-1)
4x – 4 = 16x -8
-12x = -4
x = 1/3

f(g(x)) = (4/2x) – 1 = 2/x – 1

g(f(x)) = 2[4/x – 1] = 8/x – 2

set them equal to find the solution set

[2/x – 1] = [8/x – 2]

multiply both sides by x

2 – x = 8 – 2x

x = 6