To find the inverse of the function \( f(x) = 2x + 1 \), we need to follow these steps:

1. **Replace \( f(x) \) with \( y \):**
   \[
   y = 2x + 1
   \]

2. **Swap \( x \) and \( y \):** This step helps in finding the inverse function.
   \[
   x = 2y + 1
   \]

3. **Solve for \( y \):** Isolate \( y \) on one side of the equation.
   \[
   x - 1 = 2y
   \]
   \[
   y = \frac{x - 1}{2}
   \]

4. **Replace \( y \) with \( f^{-1}(x) \):** This gives us the inverse function.
   \[
   f^{-1}(x) = \frac{x - 1}{2}
   \]

So, the inverse function is \( f^{-1}(x) = \frac{x - 1}{2} \).

Would you like more details, or do you have any other questions?

Here are some related questions:

1. How do you verify that two functions are inverses of each other?
2. What is the domain and range of the inverse function \( f^{-1}(x) = \frac{x - 1}{2} \)?
3. How does finding the inverse of a quadratic function differ from a linear function?
4. Can every function have an inverse? Why or why not?
5. How does the graph of a function compare to its inverse?

**Tip:** When finding the inverse function, always ensure that the original function is one-to-one, meaning it passes the Horizontal Line Test.

This tutorial guides you through the steps to find the inverse of the linear function f(x) = 2x + 1. Learn how to swap variables, solve for y, and verify the inverse. Perfect for students in Grades 8-10.

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