To find the inverse of the function $f(x) = \frac{3}{2}x + 3$, follow these steps:

Step 1: Replace $f(x)$ with $y$

We start by writing the equation as: $y = \frac{3}{2}x + 3$

Step 2: Swap $x$ and $y$

To find the inverse, switch $x$ and $y$: $x = \frac{3}{2}y + 3$

Step 3: Solve for $y$

Now, solve for $y$:

1. Subtract 3 from both sides: $x - 3 = \frac{3}{2}y$

2. Multiply both sides by the reciprocal of $\frac{3}{2}$, which is $\frac{2}{3}$, to isolate $y$: $y = \frac{2}{3}(x - 3)$

Step 4: Write the inverse function

The inverse function is: $f^{-1}(x) = \frac{2}{3}(x - 3)$

This is the inverse function in slope-intercept form $y = mx + b$, where:

• The slope $m = \frac{2}{3}$
• The y-intercept $b = -2$

Final Answer:

$f^{-1}(x) = \frac{2}{3}x - 2$

Would you like more details or have any questions? Here are five follow-up questions you might explore:

1. How do you verify that a function and its inverse are correct?
2. What is the slope of the original function, and how does it relate to the slope of the inverse?
3. How does switching $x$ and $y$ help in finding the inverse function?
4. Can you graph both the original function and its inverse to visualize their relationship?
5. How does the domain and range of the function change with its inverse?

Tip: Always check the inverse function by plugging it back into the original function to ensure they cancel each other out and return $x$.