Understanding Functions That Are Inverses of Themselves: A Comprehensive Guide

Functions that serve as their own inverses are known as involutions. The most common example of such a function is fx -x, which reflects points across the origin. Applying this function twice yields the original value, making it an involution.

Graph of the Function

To visualize the graph of fx -x:

The line passes through the origin (0,0). It has a slope of -1, meaning for every unit moved to the right along the x-axis, you move down one unit along the y-axis. Key characteristics include a straight line, symmetry with respect to the origin, and the line of reflection bisects the angle between the axes.

Here's how this graph is represented:

Graph of fx -x

Other Examples

Beyond fx -x, other examples include:

Absolute Value Function

The absolute value function is given by fx |x|, but its inverse is not the same. However, a different expression like fx x if x and fx -x if x > 0 can serve as its own inverse.

Identity Function

The identity function fx x is trivially its own inverse since applying it twice yields the same value.

Graph Representation

Here's a representation of the graph of fx -x:

Representation of the graph of fx -x

Summary

In summary, the graph of a function that is its own inverse, such as fx -x, is a straight line through the origin with a slope of -1. Such functions must be one-to-one, meaning they cut each horizontal and vertical line at most once, and be symmetric with respect to the line y x.

Examples of Involutions

Some non-trivial examples of involutions include:

y 1/x for x ≠ 0: This function is analytic and serves as its own inverse. y 1 - 1/2 (1 - x) for x and y 1 2 (1 - x) for x 1: This function is continuous and serves as its own inverse.

Verification Through Reflection

The graph of the inverse of a function is the reflection of the graph of this function around the line y x. This can be verified by reflecting the graphs of fx x and gx -x in the line y x, showing that the reflected graphs are indeed the same as the original.