When working with inverse functions, there are some common mistakes that can cause confusion. Here’s a simple guide to avoid these problems:

1. Not checking if a function has an inverse: One big mistake is thinking every function can be flipped. For a function to have an inverse, it must be one-to-one. This means that each output (or $y$ value) should come from only one input (or $x$ value). If a horizontal line crosses the graph more than once, then the function does not have an inverse. To avoid this mistake, take a good look at the graph or the function’s equation.

2. Swapping inputs and outputs incorrectly: Finding an inverse function means switching $x$ and $y$. A common error is not swapping them the right way. For example, if you have the function $y = 2x + 3$, you should first swap it to get $x = 2y + 3$. Make sure to do this swap carefully, then solve for $y$ to find the correct inverse.

3. Not understanding domain and range: The range of one function becomes the domain of its inverse and vice versa. If you forget this, you might draw the wrong conclusions about the inverse. For example, the function $f(x) = \sqrt{x}$ can only take numbers from $0$ to infinity ($[0, \infty)$). So, its inverse, $f^{-1}(x) = x^2$, should also have a proper domain. Always check and use the correct domain and range.

4. Forgetting to check the inverse: After you find the inverse function, it’s important to check if it’s right. You can do this by seeing if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. Many people skip this step, which can lead to mistakes. To avoid problems later, always plug the values back in to see if both equal $x$.

By being careful and paying attention to these common mistakes, you can make understanding inverse functions much easier. Knowing the basic ideas and being thorough in each step will help you succeed with inverse functions!