I'm introducing a new series in this post, called "Questions Prompted by Student Mistakes." I'm deliberately going to write these in more detail that is necessary so that non-mathematicians can read them.
On a quiz I graded recently, I saw the following:
\[ \cos\left(\frac{2}{x}\right) = \frac{\cos 2}{\cos x} = \frac{2}{x} \]because \( \cos \) totally works like that. So this prompted the question, is there a function \( f: \mathbb{R} \to \mathbb{R} \) such that \( \frac{f(a)}{f(b)} = \frac{a}{b} \) for all real numbers \( a,b \) with \( b \neq 0 \)?
Clearly we can do this for nontrivial linear functions. Every nontrivial linear function \( f: \mathbb{R} \to \mathbb{R} \) is of the form \( f(x) = \alpha x \) where \( \alpha \) is your favorite non-zero constant. We have that
\[ \frac{f(a)}{f(b)} = \frac{\alpha a}{\alpha b} = \frac{a}{b} \]So the question is, then, are linear functions the only ones for which this property holds? Click the button below to see the answer.