Problem of the Week #29: Monday July 24th, 2023 As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
Let’s consider all real valued functions whose inverse is the function itself. This means:
Since the graphs of these functions are their own mirror images across the line y=x, let’s define these functions as mirror functions.
a). Find all linear mirror functions.
b). Find a non-linear mirror function.
c). Find all mirror functions made up of two linear pieces and have a domain of all real numbers.
d). If we start with the unit circle x2 + y2 = 1, find all the ways we can remove parts of the graph of the relation to have a mirror function with the largest possible domain interval(s).
e). Find all mirror functions that are parts of a circular relation with the largest possible domain intervals, as in part d).
f). Find a mirror function made up of an exponential piece and a logarithmic piece and have a domain of all real numbers.
g). Sketch a parabola that is a mirror relation (where the graph of the relation is its own mirror image across the line y=x) and find its equation.
h). Sketch an ellipse that is a mirror relation and find its equation.
i). Share your own problem inspired by this one.
j). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.