To find the inverse of a function, you need to follow these steps. The inverse of a function \( f(x) \) is denoted by \( f^{-1}(x) \). The idea is that if \( y = f(x) \), then \( x = f^{-1}(y) \).

### **Steps to Find the Inverse of a Function**

1. **Start with the function equation:** Begin with the equation \( y = f(x) \).

2. **Swap the variables:** Replace \( y \) with \( x \) and \( x \) with \( y \). This reflects the idea that in the inverse function, the roles of the input and output are reversed.

   \[
   x = f(y)
   \]

3. **Solve for \( y \):** Manipulate the equation algebraically to solve for \( y \) in terms of \( x \). The expression you obtain will be the inverse function \( f^{-1}(x) \).

4. **Express the inverse function:** Once you have solved for \( y \), rewrite the equation as \( y = f^{-1}(x) \).

5. **Verify the inverse (optional):** To ensure correctness, you can verify by checking that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

### **Example:**

Find the inverse of the function \( f(x) = 2x + 3 \).

1. **Start with the function equation:**

   \[
   y = 2x + 3
   \]

2. **Swap the variables:**

   \[
   x = 2y + 3
   \]

3. **Solve for \( y \):**

   \[
   x - 3 = 2y
   \]

   \[
   y = \frac{x - 3}{2}
   \]

4. **Express the inverse function:**

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

### **Verification:**

To verify, substitute \( f^{-1}(x) \) into \( f(x) \):

- \( f(f^{-1}(x)) = f\left(\frac{x - 3}{2}\right) = 2\left(\frac{x - 3}{2}\right) + 3 = x - 3 + 3 = x \)
- \( f^{-1}(f(x)) = f^{-1}(2x + 3) = \frac{(2x + 3) - 3}{2} = \frac{2x}{2} = x \)

Both checks confirm that the inverse is correct.

### **Things to Note:**

- Not all functions have inverses. A function must be *one-to-one* (bijective) to have an inverse, which means it passes both the horizontal line test and vertical line test.
- The inverse of a function, if it exists, will reverse the input-output relationship of the original function.

Would you like to try finding the inverse of another function or explore more about one-to-one functions?

### **Related Questions:**
1. How do you determine if a function is one-to-one?
2. What is the significance of the horizontal line test in determining inverses?
3. Can all quadratic functions have inverses?
4. How do you find the inverse of a function involving square roots?
5. What is the relationship between the graph of a function and its inverse?

**Tip:** Always check that the original function is one-to-one before attempting to find its inverse, as not all functions have inverses that are also functions.

Learn how to find the inverse of a function with this detailed step-by-step guide. Understand the concept of inverse functions, how to swap variables, solve algebraically, and verify the correctness. Suitable for high school students studying algebra and functions.

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