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The problem asks to find the formula for \( f(x) \) given the following information:

\[
f(g(x)) = \frac{4x - 5}{6x + 1}, \quad x \neq -\frac{1}{6}
\]
and
\[
g(x) = 2x - 3.
\]

To solve for \( f(x) \), follow these steps:

### Step 1: Substitute \( g(x) = 2x - 3 \) into \( f(g(x)) \).
We know that \( f(g(x)) = \frac{4x - 5}{6x + 1} \), and we also know \( g(x) = 2x - 3 \). Let’s substitute \( g(x) = 2x - 3 \) into the equation.

Let \( u = g(x) = 2x - 3 \), then:
\[
f(u) = \frac{4x - 5}{6x + 1}.
\]

Now, solve for \( x \) in terms of \( u \):
\[
u = 2x - 3 \implies x = \frac{u + 3}{2}.
\]

### Step 2: Substitute \( x = \frac{u + 3}{2} \) into the formula for \( f(g(x)) \).
Substitute \( x = \frac{u + 3}{2} \) into the equation \( f(g(x)) = \frac{4x - 5}{6x + 1} \):

\[
f(u) = \frac{4\left( \frac{u + 3}{2} \right) - 5}{6\left( \frac{u + 3}{2} \right) + 1}.
\]

### Step 3: Simplify the expression.
First, simplify the numerator:
\[
4\left( \frac{u + 3}{2} \right) - 5 = 2(u + 3) - 5 = 2u + 6 - 5 = 2u + 1.
\]

Next, simplify the denominator:
\[
6\left( \frac{u + 3}{2} \right) + 1 = 3(u + 3) + 1 = 3u + 9 + 1 = 3u + 10.
\]

So, we now have:
\[
f(u) = \frac{2u + 1}{3u + 10}.
\]

### Step 4: Final answer.
Since \( u = g(x) \), the formula for \( f(x) \) is:

\[
f(x) = \frac{2x + 1}{3x + 10}.
\]

---

Would you like any further explanation on the steps or details?

Here are five related questions to consider:
1. How do we verify the result for \( f(x) \) by substitution back into \( f(g(x)) \)?
2. How does the domain restriction \( x \neq -\frac{1}{6} \) affect the solution process?
3. What is the importance of finding \( g(x) \) in terms of \( x \)?
4. How can we use composite functions to solve similar problems?
5. What are other methods to solve for functions in composite function problems?

**Tip:** Always double-check your substitutions in composite function problems by working backward to ensure consistency.

Jika diketahui f(g(x)) = (4x - 5) / (6x + 1), x ≠ -1/6 dan g(x) = 2x - 3, rumus dari f(x) adalah ...

Learn how to find the formula for f(x) given the composite function f(g(x)) = (4x - 5) / (6x + 1) and g(x) = 2x - 3. This detailed solution walks you through the steps of substitution and algebraic manipulation. Ideal for students in Grades 10-12.

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