How can you find the inverse of a function?

To find the inverse of a function, you essentially reverse the roles of the input (independent variable) and output (dependent variable) and then solve for the new output variable. Here's a step-by-step guide:

### Step 1: Write the function in terms of \( y \)
Given a function \( f(x) \), start by expressing it as:
\[
y = f(x)
\]
This means that \( y \) is the output when \( x \) is the input.

### Step 2: Swap \( x \) and \( y \)
To find the inverse, switch the roles of \( x \) and \( y \). Replace \( y \) with \( x \) and \( x \) with \( y \):
\[
x = f(y)
\]

### Step 3: Solve for \( y \)
Now, solve this equation for \( y \). The solution you obtain will be the expression for the inverse function, which we'll denote as \( f^{-1}(x) \).

### Step 4: Verify the Inverse
To ensure your inverse is correct, you can perform the following checks:
1. **Check if \( f(f^{-1}(x)) = x \)**: Substitute \( f^{-1}(x) \) into the original function and see if you get back \( x \).
2. **Check if \( f^{-1}(f(x)) = x \)**: Substitute \( f(x) \) into the inverse function and see if you get back \( x \).

### Example
Let's find the inverse of the function \( f(x) = 2x + 3 \).

1. **Express the function**:
   \[
   y = 2x + 3
   \]

2. **Swap \( x \) and \( y \)**:
   \[
   x = 2y + 3
   \]

3. **Solve for \( y \)**:
   \[
   x - 3 = 2y
   \]
   \[
   y = \frac{x - 3}{2}
   \]

   So, the inverse function is:
   \[
   f^{-1}(x) = \frac{x - 3}{2}
   \]

4. **Verify the inverse**:
   - Check \( f(f^{-1}(x)) \):
     \[
     f\left(\frac{x - 3}{2}\right) = 2\left(\frac{x - 3}{2}\right) + 3 = x - 3 + 3 = x
     \]
   - Check \( f^{-1}(f(x)) \):
     \[
     f^{-1}(2x + 3) = \frac{2x + 3 - 3}{2} = \frac{2x}{2} = x
     \]
   Both checks confirm that the inverse is correct.

Would you like more details or have any questions?

### Related Questions:
1. How do you find the inverse of a quadratic function?
2. Can every function have an inverse? Why or why not?
3. What is the geometric interpretation of the inverse function on a graph?
4. How can you find the inverse of a function with a square root?
5. How do you verify if two functions are inverses of each other?

**Tip:** Remember that a function must be one-to-one (each \( x \) maps to exactly one \( y \) and each \( y \) maps to exactly one \( x \)) for its inverse to be a function.

Learn how to find the inverse of a function with our detailed step-by-step guide. Understand the concept of inverse functions, swap roles of input and output variables, solve for the new output variable, and verify the inverse. Suitable for Grades 10-12 students and anyone learning about functions and their inverses.

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