Equality of Mappings/Unequal/Examples/Exponential Functions

Example of Mappings which are Unequal
Let $\theta: \R \to \R$ be the mapping from the set of real numbers to itself defined as:
 * $\forall x \in \R: \map \theta x := e^x$

where $e^x$ denotes the real exponential function.

Let $\phi: \R \to \R_{>0}$ be the mapping from the set of real numbers to the set of (strictly) positive real numbers defined as:
 * $\forall x \in \R: \map \theta x := e^x$

Then:
 * $\phi \ne \theta$

because, while $\Dom \theta = \Dom \phi$, and $\forall x \in \R: \map \theta x = \map \phi x$, they have different codomains:
 * $\Cdm \theta = \R \ne \R_{>0} = \Cdm \phi$