A calculus of the absurd

4.12.2 Solutions for problems in applying functions to inequalities

If we have a function \(f(x)\) which is decreasing (i.e. as the inputs become larger, the outputs decrease) then to apply it to both sides of an inequality, it is necessary to “flip” the inequality. For example, if we pick the reciprocal function3939 Lucky reciprocal function to be \(f(x)\), (i.e. we set \(f(x)=\frac {1}{x}\)), we might want to apply \(f(x)\) to both sides of

\begin{equation} a > b \end{equation}

To ensure that we still have a valid inequality, we end up with

\begin{equation} \frac {1}{a} < \frac {1}{b} \end{equation}