if $f(x)=\frac{4}{x - 1}$ and $g(x)=2x$, then the solution set of $f(g(x)) = g(f(x))$ is

Explanation:

Step1: Find $f(g(x))$

Substitute $g(x)=2x$ into $f(x)$: $f(g(x))=\frac{4}{2x - 1}$

Step2: Find $g(f(x))$

Substitute $f(x)=\frac{4}{x - 1}$ into $g(x)$: $g(f(x))=2\times\frac{4}{x - 1}=\frac{8}{x - 1}$

Step3: Set $f(g(x)) = g(f(x))$

$\frac{4}{2x - 1}=\frac{8}{x - 1}$ Cross - multiply: $4(x - 1)=8(2x - 1)$

Step4: Expand and solve for $x$

Expand: $4x-4 = 16x-8$ Move $x$ terms to one side: $16x - 4x=8 - 4$ $12x = 4$ $x=\frac{1}{3}$

Answer:

$\left{\frac{1}{3}\right}$