👉 In mathematics, a vacantly defined function is an undefined mathematical function that has no value. It can be represented by the symbol "$\varnothing$" or "∅". This notation signifies that there is no value associated with it in terms of its inputs and outputs. For example, consider the following function: $$f(x) = x^2 - 1$$ This function is vacantly defined because there are no values that satisfy this equation for all real numbers $x$. The expression