Creating a new symbol in LaTeX

Question

I would like to create a symbol for a function in LaTeX, like this:

Can someone help me?

It would have to look like this:

Answer 1

I provide, at no cost to you, \froogle...works in all math styles.

\documentclass{article}
\usepackage{graphicx,scalerel}
\newcommand\froogle{\scalerel*{\rotatebox{90}{\raisebox{1pt}{%
  \scalebox{1}[.7]{$\varsigma$}}\kern-1.1pt\scalebox{.7}[1]{o}}}{Xg}}
\begin{document}
$\froogle(x) + A_{\froogle_{(1 + \froogle)}}$
\end{document}

If you change fonts, the parameters may need tweaking. Here, I redo for txfonts:

\documentclass{article}
\usepackage{graphicx,scalerel,txfonts}
\newcommand\froogle{\scalerel*{\rotatebox{90}{\raisebox{1.3pt}{%
  \scalebox{.85}[.7]{$\varsigma$}}\kern-1.2pt\scalebox{.85}[1]{o}}}{Xg}}
\begin{document}
$\froogle(x) + A_{\froogle_{(1 + \froogle)}}$
\end{document}

Answer 2

I think that you can find three symbols that are close to one you desire. The $\varrho$ from Fran's answer, and two \mathfrak characters (the g and d, transformed).

The code is here.

\documentclass[11pt, border=1cm]{standalone}
\usepackage{amssymb}
\usepackage{graphicx}
\begin{document}
$x\mapsto\varrho(x)$ 
\qquad
$x\mapsto\raisebox{-.2ex}{\rotatebox[origin=c]{180}{$\mathfrak{d}$}}(x)$
\qquad
$x\mapsto\raisebox{.35ex}{\reflectbox{$\mathfrak{g}$}}(x)$
\end{document}