Editing Involution (mathematics) (section)

=== Real-valued functions ===

The [[Graph_of_a_function|graph]] of an involution (on the real numbers) is [[Reflection_symmetry|symmetric]] across the line {{math|1=''y'' = ''x''}}. This is due to the fact that the inverse of any ''general'' function will be its reflection over the line {{math|1=''y'' = ''x''}}. This can be seen by "swapping" {{mvar|x}} with {{mvar|y}}. If, in particular, the function is an ''involution'', then its graph is its own reflection. 

Some basic examples of involutions include the functions
<math display="block">\begin{alignat}{1}
f(x) &= a-x \; , \\
f(x) &= \frac{b}{x-a}+a
\end{alignat}</math>Besides, we can construct an involution by wrapping an involution {{mvar|g}} in a bijection {{mvar|h}} and its inverse (<math>h^{-1} \circ  g \circ h</math>). For instance :<math display="block">\begin{alignat}{2}
f(x) &= \sqrt{1 - x^2} \quad\textrm{on}\; [0;1] & \bigl(g(x) = 1-x \quad\textrm{and}\quad h(x) = x^2\bigr), \\
f(x) &= \ln\left(\frac {e^x+1}{e^x-1}\right) & \bigl(g(x) = \frac{x+1}{x-1}=\frac{2}{x-1}+1 \quad\textrm{and}\quad h(x) = e^x\bigr)  \\
\end{alignat}</math>