Diketahui f(3x)  = x dan g = (x-1/x-2) = 2x . Jika fungsi f dan g mempunyai invers, maka (fog)^-1 (5)

To solve the problem, we need to find the inverse of the composition \((f \circ g)^{-1}\). Given the information:

1. \( f(3x) = x \)
2. \( g(x) = \frac{x-1}{x-2} = 2x \) (this statement seems off, let's assume it means \( g(x) = \frac{x-1}{x-2} \))
   
### Step 1: Finding the inverse of \(f(x)\)
Given \( f(3x) = x \). To find \( f^{-1}(x) \), let’s rewrite it in terms of \(x\):

\[ f(3x) = x \implies y = 3x \implies x = \frac{y}{3} \implies f(x) = \frac{x}{3}. \]

Now, finding the inverse:

\[ f(x) = \frac{x}{3} \implies f^{-1}(x) = 3x. \]

### Step 2: Finding the inverse of \(g(x)\)
Given \( g(x) = \frac{x-1}{x-2} \). To find \( g^{-1}(x) \):

Set \( y = \frac{x-1}{x-2} \). Cross-multiply to solve for \(x\):

\[
y(x-2) = x-1 \implies yx - 2y = x - 1 \implies yx - x = 2y - 1 \implies x(y-1) = 2y - 1 \implies x = \frac{2y-1}{y-1}.
\]

So,

\[
g^{-1}(x) = \frac{2x-1}{x-1}.
\]

### Step 3: Finding \((f \circ g)^{-1}(5)\)
First, find \(f \circ g(x)\):

\[
f(g(x)) = f\left(\frac{x-1}{x-2}\right).
\]

Substituting \(g(x)\) into \(f\):

\[
f\left(\frac{x-1}{x-2}\right) = \frac{\frac{x-1}{x-2}}{3} = \frac{x-1}{3(x-2)}.
\]

The inverse of \(f \circ g\) is needed at \(5\):

Let \(y = \frac{x-1}{3(x-2)}\). To find the inverse, solve for \(x\):

\[
5 = \frac{x-1}{3(x-2)}.
\]

Cross-multiplying gives:

\[
5 \cdot 3(x-2) = x-1 \implies 15x - 30 = x - 1 \implies 14x = 29 \implies x = \frac{29}{14}.
\]

Thus, \((f \circ g)^{-1}(5) = \frac{29}{14}\).

Would you like further clarification on any step, or explore related questions?

### Related Questions:
1. How do you find the composition of functions in general?
2.

Explore how to solve the problem (f ∘ g)^{-1}(5) using function composition and inverse functions. This problem involves determining the inverse of the composition of two functions, f and g, given specific functional relationships. Suitable for advanced mathematics students and those interested in function theory.

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