Example: $f:\mathbb{R}\to \mathbb{R}$ given by $f(x)=\frac{1}{x^2+1}$. What is $f^{-1}(\frac{1}{3})$? What is $f^{-1}(\{\frac{1}{3}\})$?
My guess is that: either $f^{-1}(\frac{1}{3})$ is undefined as it has two target values and $f^{-1}(\{\frac{1}{3}\})=\{-\sqrt{2}, \sqrt{2}\}$; or both are undefined.
Although, strictly speaking, $f^{-1}(a)$ is undefined if $f$ has no inverse, it is usual to use the expression $f^{-1}(a)$ as meaning the same thing as $f^{-1}\bigl(\{a\}\bigr)$.