Percentage decrease?
An interesting function problem
Problem and Solution
This functional equation may look simple at first glance, but to solve it you will need to spot one beautiful step. I believe that doing maths is the only way to truly understand it, so I really suggest trying this problem before looking at the solution development below.
Solution
This problem requires us to start by performing a little bit of rearranging. First, we take a one out of the left-hand side:
Then we move this to the right and continue rearranging to get a formula for f(x):
Then, there is the beautiful step. We plug it into itself “x” times. Giving:
We are able to this if afterwords it can be solved to form a continuous function, which is the real part of what is known as a Holomorphic function. But that is off topic for this article.
Next, we are able to take the denominators of most of the fractions and cancel them out with the previous numerator! Which takes the long product and turns it into:
Which is the solution to our problem, as it gives an infinite family of solutions for f(x) depending on our value of f(1). How amazing!
As an additional challenge, why not try to prove that this solution also works for non-integers.
Conclusion
I believe that everyone should be able to enjoy maths, and getting started solving problems is a great way to do this. So, if you did this problem using a different method, comment on how you did it. I would love to know and as always,
Have fun and never stop puzzling.
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