Surjective Restriction of Real Exponential Function

Theorem

Let $\exp: \R \to \R$ be the exponential function:

$\map \exp x = e^x$

Then the restriction of the codomain of $\exp$ to the strictly positive real numbers:

$\exp: \R \to \R_{>0}$

is a surjective restriction.

Hence:

$\exp: \R \to \R_{>0}$

is a bijection.

Proof

We have Exponential on Real Numbers is Injection.

Let $y \in \R_{> 0}$.

Then $\exists x \in \R: x = \map \ln y$

That is:

$\exp x = y$

and so $\exp: \R \to \R_{>0}$ is a surjection.

Hence the result.

$\blacksquare$

Sources

• 1959: E.M. Patterson: Topology (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 9$. Functions: Example $2$
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.3$. Injective, surjective, bijective; inverse mappings: Example $48$