What is an inverse function?

An inverse function reverses the effect of the original function, swapping the input and output values.

In more detail, if you have a function \( f(x) \) that takes an input \( x \) and gives an output \( y \), the inverse function, denoted as \( f^{-1}(x) \), will take \( y \) as its input and return \( x \) as its output. Essentially, if \( f(a) = b \), then \( f^{-1}(b) = a \). This relationship means that applying a function and then its inverse will get you back to your starting point: \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

To find the inverse of a function, you typically swap the \( x \) and \( y \) in the equation and then solve for \( y \). For example, if you have the function \( f(x) = 2x + 3 \), you would start by writing it as \( y = 2x + 3 \). Then, swap \( x \) and \( y \) to get \( x = 2y + 3 \). Finally, solve for \( y \) to find the inverse: \( y = \frac{x - 3}{2} \), so \( f^{-1}(x) = \frac{x - 3}{2} \).

It's important to note that not all functions have inverses. For a function to have an inverse, it must be one-to-one, meaning that each output is produced by exactly one input. This ensures that the inverse function will also be a function, passing the vertical line test (each vertical line intersects the graph at most once).

Understanding inverse functions is crucial in many areas of mathematics, including solving equations and analysing real-world problems where reversing processes is necessary.