Describe the process to find the inverse of a function.

Finding the inverse of a function involves a step-by-step process to reverse the roles of inputs and outputs. Here's a detailed description of the process:

### Step-by-Step Process to Find the Inverse of a Function

1. **Write the function as \( y = f(x) \)**: 
   Begin by expressing the given function in the form \( y = f(x) \). This step makes it easier to work with the function algebraically.

2. **Interchange \( x \) and \( y \)**: 
   To find the inverse, swap the roles of \( x \) and \( y \). This reflects the idea that for an inverse, the output becomes the input, and the input becomes the output. So the new equation will be \( x = f(y) \).

3. **Solve for \( y \)**: 
   Solve the new equation for \( y \). The resulting expression for \( y \) will represent the inverse function \( f^{-1}(x) \).

4. **Express the inverse function**: 
   After solving for \( y \), replace \( y \) with \( f^{-1}(x) \). This final expression is the inverse of the original function.

5. **Verify the result (optional)**: 
   To check if your inverse function is correct, you can verify by confirming that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

### Example:
Let's consider a function \( f(x) = 2x + 3 \).

1. Write it as \( y = 2x + 3 \).
2. Swap \( x \) and \( y \): \( x = 2y + 3 \).
3. Solve for \( y \):
   \[
   x - 3 = 2y \quad \Rightarrow \quad y = \frac{x - 3}{2}
   \]
4. Therefore, the inverse function is \( f^{-1}(x) = \frac{x - 3}{2} \).
5. Verify by checking:
   \[
   f(f^{-1}(x)) = 2\left(\frac{x - 3}{2}\right) + 3 = x
   \]
   \[
   f^{-1}(f(x)) = \frac{2x + 3 - 3}{2} = x
   \]

This confirms that the inverse is correct.

Would you like more details or have any questions?

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### Related Questions:
1. What are the conditions for a function to have an inverse?
2. How do you find the inverse of a quadratic function?
3. What happens if a function is not one-to-one? Can it still have an inverse?
4. How can you graphically verify the inverse of a function?
5. What is the relationship between a function and its inverse?

**Tip**: A function must be one-to-one (injective) to have an inverse, which means it passes the horizontal line test (each output corresponds to exactly one input).

Learn how to find the inverse of a function with this detailed step-by-step guide. This tutorial covers key algebraic concepts such as swapping variables, solving for y, and verifying the inverse. Suitable for high school students in Grades 9-12.

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