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).