Preimage of Subset under Mapping/Examples/Preimage of 0 under x^2-x-2

Example of Preimage of Subset under Mapping

Let $f: \R \to \R$ be the mapping defined as:

$\forall x \in \R: \map f x = x^2 - x - 2$

The preimage of the singleton $\set 0$ is:

$f^{-1} \sqbrk {\set 0} = \set {-1, 2}$

which is the set of roots of $f$.

Proof

\(\ds \map f x\)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds x^2 - x - 2\)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds \paren {x - 2} \paren {x + 1}\)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds 2 \text { or } x = -1\)

Hence the result.

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 6$: Functions