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The given function is:

\[
f(x) = \frac{a}{m(x + k)}
\]

We are asked to find the inverse \( f^{-1}(x) \). To do that, follow these steps:

### Step 1: Set \( y = f(x) \)
\[
y = \frac{a}{m(x + k)}
\]

### Step 2: Solve for \( x \)
We want to express \( x \) in terms of \( y \).

1. Multiply both sides by \( m(x + k) \) to get rid of the denominator:
   \[
   y \cdot m(x + k) = a
   \]

2. Divide both sides by \( y \):
   \[
   m(x + k) = \frac{a}{y}
   \]

3. Now, divide both sides by \( m \):
   \[
   x + k = \frac{a}{my}
   \]

4. Subtract \( k \) from both sides:
   \[
   x = \frac{a}{my} - k
   \]

### Step 3: Replace \( y \) with \( x \)
Now that we've solved for \( x \), the inverse function is:
\[
f^{-1}(x) = \frac{a}{mx} - k
\]

Thus, the inverse of the function \( f(x) = \frac{a}{m(x + k)} \) is:

\[
f^{-1}(x) = \frac{a}{mx} - k
\]

Would you like to go over the domain, range, or sketch the graph? Feel free to ask for any further clarifications!

Here are five related questions to expand the discussion:
1. What is the domain and range of \( f(x) \) and its inverse \( f^{-1}(x) \)?
2. How does the value of \( k \) affect the graph of \( f(x) \) and its inverse?
3. Is there any value of \( x \) that makes \( f(x) \) undefined? How would you find that?
4. What happens to the inverse function when \( m \) is negative?
5. Can you sketch the graph of both \( f(x) \) and \( f^{-1}(x) \) on the same axes to see the symmetry?

**Tip:** When finding the inverse of a function, always verify that the function is one-to-one by checking if it passes the horizontal line test.

Find the inverse of the rational function f(x) = a / [m(x + k)]. This problem involves algebraic manipulation and solving for the inverse. Suitable for Grades 10-12, this solution includes detailed steps for solving the inverse and covers important concepts such as domain, range, and graphing of rational functions.

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