To find the inverse of the function $f(x) = \frac{3}{8}x - 1$, we will follow these steps:

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

Let: $y = \frac{3}{8}x - 1$

Step 2: Swap $x$ and $y$

To find the inverse, we swap $x$ and $y$: $x = \frac{3}{8}y - 1$

Step 3: Solve for $y$

Now, we need to solve for $y$.

1. Add 1 to both sides: $x + 1 = \frac{3}{8}y$

2. Multiply both sides by $\frac{8}{3}$ to isolate $y$: $y = \frac{8}{3}(x + 1)$

Thus, the inverse function is: $f^{-1}(x) = \frac{8}{3}(x + 1)$

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Would you like a more detailed explanation of any of the steps, or do you have other questions about functions or inverses?

Related Questions:

1. How do we check if two functions are inverses of each other?
2. How would we find the inverse of a function involving quadratic terms?
3. Can a function have more than one inverse?
4. What is the significance of the inverse function in real-world applications?
5. How do you graph a function and its inverse on the same coordinate plane?

Tip: To graph a function and its inverse, reflect the graph of the function over the line $y = x$. The result is the graph of the inverse function.