To find the formula for the inverse of a function, follow these general steps:

Steps to Find the Inverse of a Function:

1. Start with the function: Let’s say you have a function $f(x)$, and you want to find its inverse. Denote the inverse function as $f^{-1}(x)$.

2. Replace the function notation: Replace $f(x)$ with $y$. So, you now have: $y = f(x)$

3. Switch $x$ and $y$: To find the inverse, you need to swap the roles of $x$ and $y$. This is because the inverse function "reverses" the roles of the input and output. After switching, you have: $x = f(y)$

4. Solve for $y$: Now, solve the equation $x = f(y)$ for $y$. The solution for $y$ will give you the inverse function $f^{-1}(x)$.

5. Express the inverse: Once you've isolated $y$, replace $y$ with $f^{-1}(x)$. This gives you the formula for the inverse function.

Example:

Let’s find the inverse of $f(x) = 2x + 3$.

1. Start with $y = f(x) = 2x + 3$.

2. Swap $x$ and $y$: $x = 2y + 3$

3. Solve for $y$: $x - 3 = 2y \quad \Rightarrow \quad y = \frac{x - 3}{2}$

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

Important Notes:

• Domain and Range: For the inverse function to exist, the original function must be bijective (one-to-one and onto). This means that for each output, there must be exactly one input (no duplicates), and for each input, there must be a corresponding output.
• Check the inverse: You can always verify that $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$ to ensure correctness.

Would you like me to go over any specific types of functions or help you with an example?

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Here are some related questions:

1. What does it mean for a function to be one-to-one (injective)?
2. How do you verify if a function has an inverse?
3. Can all functions have an inverse?
4. What is the geometric interpretation of an inverse function?
5. How do inverse functions relate to symmetry with respect to the line $y = x$?

Tip: Always check if the function is bijective (one-to-one and onto) before finding an inverse. If it’s not, the inverse won’t be a valid function.