Preimage of Subset under Mapping/Examples/Subset of Image of Square Root Function

Example of Preimage of Subset under Mapping
Let $f: \R \to \R$ be the real function defined as:
 * $\forall x \in \R: \map f x = x^2$

Let $A \subseteq \R$ be defined as:
 * $A := \closedint 4 9 = \set {x \in \R: 4 \le x \le 9}$

Then the preimage of $A$ under $f$ is:
 * $f^{-1} \sqbrk A = \closedint {-3} {-2} \cup \closedint 2 3$

Let $B \subseteq \R$ be defined as:
 * $B := \closedint {-9} {-4} = \set {x \in \R: -9 \le x \le -4}$

Then the preimage of $B$ under $f$ is:
 * $f^{-1} \sqbrk B = \O$