To find the inverse of the function \( f(x) = \frac{4x}{x+5} \), follow these steps:

### Step 1: Replace \( f(x) \) with \( y \):
\[
y = \frac{4x}{x+5}
\]

### Step 2: Swap \( y \) and \( x \):
\[
x = \frac{4y}{y+5}
\]
We want to solve for \( y \) in terms of \( x \).

### Step 3: Multiply both sides by \( (y+5) \) to eliminate the denominator:
\[
x(y + 5) = 4y
\]

### Step 4: Distribute \( x \) on the left side:
\[
xy + 5x = 4y
\]

### Step 5: Move all terms involving \( y \) to one side:
\[
xy - 4y = -5x
\]

### Step 6: Factor out \( y \) from the left side:
\[
y(x - 4) = -5x
\]

### Step 7: Solve for \( y \):
\[
y = \frac{-5x}{x - 4}
\]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{-5x}{x - 4}
\]

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Would you like further details on any step or have any other questions?

Here are five related questions you might explore:
1. How do you verify if the inverse function is correct?
2. What happens if you try to find the inverse of a different rational function, like \( f(x) = \frac{2x}{x-3} \)?
3. How would you graph both \( f(x) \) and \( f^{-1}(x) \)?
4. What is the domain and range of \( f^{-1}(x) \)?
5. Can all functions be inverted? Why or why not?

**Tip**: Always check the domain of the original function when finding its inverse, as it can affect the domain and range of the inverse function.

Discover how to find the inverse of the rational function f(x) = 4x / (x + 5) with a detailed, step-by-step guide. This problem covers core algebraic concepts like rational and inverse functions. Ideal for high school students (Grades 10-12) looking to strengthen their understanding of inverse functions.

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