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Inverse functions are functions that undo each other. In Example 1, use the equation solved for x to write the inverse of f by switching x and y. x = y − 3 — 2 y = x − 3 — 2 An inverse function can be denoted by f −1, read as “f inverse.” Because an inverse function switches the input and output values of the original function ...
An inverse function is a second function which undoes the work of the first one. In this unit we describe two methods for finding inverse functions, and we also explain that the domain of a function may need to be restricted before an inverse function can exist.
Functions that undo each other are called inverse functions. In Example 1, you can use the equation solved for x to write the inverse of f by switching the roles of x and y. f(x) 2x original function g(x) x 3.
Lecture 1 : Inverse functions One-to-one Functions A function f is one-to-one if it never takes the same value twice or f(x 1) 6=f(x 2) whenever x 1 6=x 2: Example The function f(x) = x is one to one, because if x 1 6=x 2, then f(x 1) 6=f(x 2). On the other hand the function g(x) = x2 is not a one-to-one function, because g( 1) = g(1).
Inverse Functions We are now going to consider the class of problems in which • we have a given function, that we’ll call f, and • for each number X • we wish to find a number Y obeying f(Y) = X (1) If we’re lucky, then for each real number X there is exactly one real number Y, that we’ll call f−1(X), obeying (1).
Inverse of a Function. The inverse of a one-to-one function , written −1, is the set of all ordered pairs of the form ( , ), where ( , ) belongs to . Since the inverse is formed by interchanging and , the domain of becomes the range of −1 and the range of becomes the domain of −1.
Examples of Inverse Functions. Suppose that we have a function that adds 3 to its input: f(x) = x + 3: The question is can we guess the input if we knew the output? For instance, if the output is 13, what would be the input? How about when f(x) = 29, what is x? In general, if we know y = f(x) = x + 3, what is x?
