What are 3 functions that are their own inverses? (10 points)?

Question

I know of only one function that is its own inverse: f(x) = 1/x. Please give me 2 more. Thanks.

Answer 1

f(x) = x and f(x) = -x are two such functions  

Answer 2

This type of question is a non-traditional one, which deserves quite of a unique attention.  Usually, students are taught in school the idea of inverse functions and how to find the inverse of a given function.  But there has never been anything said about a function being its own inverse.  Inverse Functions are one of several types of functions that define the association between functions, which is sort of like a functional hierarchy called composite functions.  However, the major difference between what you are learning in school and deep mathematics is that there are some intricate concepts that the school itself just doesn't cover perhaps it may be for the sake of introducing you to a brand new idea of mathematics called megamathematics.  

Now in finding functions that are their own inverses, you must know first what makes a function its own inverse.  Well, functions that are their own inverse are called self-inverse functions; that is, functions that complement themselves with respect to their domain.  Therefore, inverse functions are of the form: id^-1x = idx || N(N(x)) = x.  Self-Inverse functions involve a set X whose functional identity on X is the function itself.  F(x) = F(x)^-1 because F(F(x)^-1) = x && F(F(x))^-1 = x.  Self-Inverse functions have a their own unique domains && ranges that take on all values of x.

F(x) = 1 / x is a self-inverse function because by imaging another function such g(x) having the same identity and substituting it into the function f(x), you have f(g(x)) = x.  To find another self-inverse function, let the set X be the center of attention and think about any related algebraic function whose domain and range are identical such as the function f(x) = x || f(x) = -x.  If you draw the graph of both functions, you can see that the domain and range are identical irrespective to the sign.  That is, f(g(x)) = x for real numbers.  Therefore, the 3 self-inverse functions are f(x) = 1 / x, f(x) = x, && f(x) = -x.

J.C