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To solve this problem, we need to find the inverse function \( f^{-1}(w) \) of the given function:

\[
f(w) = \frac{3w}{5 + 3w} - 4
\]

### Step-by-Step Process to Find the Inverse

1. **Replace \( f(w) \) with \( y \):**

   \[
   y = \frac{3w}{5 + 3w} - 4
   \]

2. **Solve for \( w \) in terms of \( y \):**

   Start by isolating the fraction:

   \[
   y + 4 = \frac{3w}{5 + 3w}
   \]

   Cross-multiply to eliminate the fraction:

   \[
   (y + 4)(5 + 3w) = 3w
   \]

   Expand the equation:

   \[
   5(y + 4) + 3w(y + 4) = 3w
   \]

   Rearrange the terms to isolate \( w \):

   \[
   5y + 20 = 3w - 3wy
   \]

   Factor out \( w \) on the right side:

   \[
   5y + 20 = w(3 - 3y)
   \]

   Solve for \( w \):

   \[
   w = \frac{5y + 20}{3 - 3y}
   \]

3. **Express \( w \) as a function of \( y \):**

   Now, replace \( y \) with \( w \) to find the inverse:

   \[
   f^{-1}(w) = \frac{5w + 20}{3 - 3w}
   \]

4. **Find the logarithm that matches this form:**

   None of the logarithm options given match this exact form directly, but let's check each:

   - The first option involves a logarithm with base 3, and a transformation of terms \( \frac{5(w+4)}{3-w} \), which appears similar to our transformation above.
   
To determine which answer fits the correct inverse form, note that the expression:

\[
f^{-1}(w) = \frac{5w + 20}{3 - 3w}
\]

transforms correctly and aligns with the function whose logarithm matches most closely in the first answer:

**The correct answer is likely the first option.**

If you'd like further detailed steps or clarification, please let me know!

Elija la opción en la que se define la función inversa de la función uno a uno definida por f(w) = \frac{3w}{5 + 3w} - 4.

Learn how to find the inverse function for the given rational expression f(w) = (3w)/(5 + 3w) - 4. This problem involves key concepts of algebra and logarithms, ideal for high school students in grades 10-12.

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