What is the inverse function?

An inverse function is a function that reverses the operation done by the original function. To understand this, consider a function f that takes an input x and produces an output y, which can be written as y = f(x). The inverse function, usually denoted as f^(-1), takes y as its input and gives back the original x. In other words, if you had y = f(x), then x = f^(-1)(y).

We can say two functions f and f^(-1) are inverses if and only if both of the following conditions are satisfied for every x in the domain of f and every y in the domain of f^(-1):

1. f(f^(-1)(y)) = y 2. f^(-1)(f(x)) = x

For a function to have an inverse, it must be bijective, that means it needs to be both injective (one-to-one) and surjective (onto).

To find the inverse of a function algebraically, you typically follow these steps:

1. Start with the function equation y = f(x). 2. Swap x and y to make x the subject of the equation: x = f^(-1)(y). 3. Solve this new equation for y to get y = g(x), where g(x) is the inverse function of f(x).

Graphically, the inverse function is a reflection of the original function across the line y = x. The intersection points with this line remain the same, while points not on it are flipped over the line.