Preimage of Subset under Mapping/Examples/Preimage of -5 to -4 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 closed interval $\closedint {-5} {-4}$ is:

$f^{-1} \closedint {-2} 0 = \O$

the Empty Set

Proof

Trivially, by differentiating $x^2 - x - 2$ with respect to $x$:

$f' = 2 x - 1$

and equating $f'$ to $0$, the minimum of $\Img f$ is seen to occur at $\map f {\dfrac 1 2} = -\dfrac 9 4$.

We see that $\closedint {-5} {-4}$ is well outside the image of $f$.

Hence the result.

$\blacksquare$

Sources

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