Building blocks

## Solution

Sketch the following functions and determine which of them are self-inverse.

What do the graphs of the self-inverse functions have in common?

We have sketched the four self-inverse functions on the same set of axes.

We remember that the graphs of any function and its inverse are reflections of one another in the line $y=x$. So it is no surprise that the graph of each self-inverse function is a reflection of itself in $y=x$.

Can you match up the four functions with their graphs?

Sketch and find equations for some other self-inverse functions.

By thinking about the reflection property we can see that any straight line parallel to the purple line in the graph above will represent a self-inverse function. Can you write down a generalised form of the equation?

Also, any translation of $y=\dfrac{1}{x}$ will be self-inverse so long as the origin is transformed to a point on the line $y=x$. What about stretches, reflections and other transformations?

Can you find any other families of self-inverse functions?