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My text uses a different font to represent higher order functions that give the even and odd parts of a function. How do I write these in Latex?

picture from textbook

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    Welcome to TeX.SE! Can you show us some tex code you have tried so far? – Mensch Aug 21 at 3:10
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I assume your question is about how to write "Ev" and "Od" using math-calligraphic lettering.

For the uppercase letters "E" and "O", just input \mathcal{E} and \mathcal{O}, respectively.

Computer Modern, which is the default text and math font family in virtually all TeX installations, does not provide true calligraphic lowercase letters. In fact, very few math fonts offer true lowercase calligraphic letters. The following is a table with 11 possibilities; note that it's necessary to use LuaLaTeX for the \usepackage{Unicode-math} and various \setmathfont directives.

enter image description here

% !TEX TS-program = lualatex
\documentclass{article}
\newcommand\test{$\mathcal{Ev}\{x(t)\} \quad \mathcal{Od}\{x(t)\}$}

\usepackage{unicode-math}
%% Load various math fonts:
\setmathfont{Latin Modern Math}[version=LM]
\setmathfont{Stix Math}[version=Stix]
\setmathfont{XITS Math}[version=XITS]
\setmathfont{Stix Two Math}[version=Stix2]
\setmathfont{Cambria Math}[version=Cambria]
\setmathfont{Asana Math}[version=Asana]
\setmathfont{Asana Math}[Alternate,version=AsanaAlt]
\setmathfont{TeX Gyre Pagella Math}[version=Pagella]
\setmathfont{TeX Gyre Termes Math}[version=Termes]
\setmathfont{TeX Gyre DejaVu Math}[version=DejaVu]
\setmathfont{Libertinus Math}[version=Libertinus]

\begin{document}
\renewcommand\arraystretch{1.5}
\begin{tabular}{@{}ll@{}}
Latin Modern   & \mathversion{LM}      \test \\
Stix           & \mathversion{Stix}    \test \\
XITS           & \mathversion{XITS}    \test \\
Stix Two       & \mathversion{Stix2}   \test \\
Cambria        & \mathversion{Cambria} \test \\
Asana          & \mathversion{Asana}   \test \\
Asana Alt      & \mathversion{AsanaAlt}\test \\
Pagella        & \mathversion{Pagella} \test \\
Termes         & \mathversion{Termes}  \test \\
DejaVu         & \mathversion{DejaVu}  \test \\
Libertinus Math& \mathversion{Libertinus} \test
\end{tabular}
\end{document} 

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