Inverse function

 Inverse Functions

Let’s say you have a function f. You present it with an input x; provided that x is in the domain of f, you get back an output, which we call f(x). Now we try to do things all backward and ask this question: if you pick a number y, what input can you give to f in order to get back y as your output? Here’s how to state the problem in math-speak: given a number y, what x in the domain of f satisfies f(x) = y? The first thing to notice is that y has to be in the range of f. Otherwise, by definition there are no values of x such that f(x) = y. There would be nothing in the domain that f would transform into y, since the range is all the possible outputs. On the other hand, if y is in the range, there might be many values that work. For example, if f(x) = x2 (with domain R), and we ask what value of x transforms into 64, there are obviously two values of x: 8 and −8. On the other hand, if g(x) = x3, and we ask the same question, there’s only one value of x, which is 4. The same would be true for any number we give to g to transform, because any number has only one (real) cube root. So, here’s the situation: we’re given a function f, and we pick y in the range of f. Ideally, there will be exactly one value of x which satisfies f(x) = y. If this is true for every value of y in the range, then we can define a new function which reverses the transformation. Starting with the output y, the new function finds the one and only input x which leads to the output. The new function is called the inverse function of f, and is written as f−1. Here’s a summary of the situation in mathematical language:
1. Start with a function f such that for any y in the range of f, there is exactly one number x such that f(x) = y. That is, different inputs give different outputs. Now we will define the inverse function f−1. 2. The domain of f−1 is the same as the range of f. 3. The range of f−1 is the same as the domain of f. 4. The value of f−1(y) is the number x such that f(x) = y. So, if f(x) = y, then f−1(y) = x. The transformation f−1 acts like an undo button for f: if you start with x and transform it into y using the function f, then you can undo the effect of the transformation by using the inverse function f−1 on y to get x back.