Label of enumerate as a math environment with number "1" in subscript

Use \bgroup and \egroup:

\documentclass{article}
\usepackage{enumerate}
\begin{document}
\begin{enumerate}[$f_\bgroup1\egroup:x\mapsto$]
\item $x$
\item $x^2$
\item $x^3$
\item $x^4$
\item $x^5$
\item $x^6$
\item $x^7$
\item $x^8$
\item $x^9$
\item $x^{10}$
\end{enumerate}
\end{document}

You need the enumitem package, since it allows more freedom for specifying the appropriate label and formatting:

\documentclass{article}

\usepackage{enumitem}

\begin{document}

\begin{enumerate}[label = {$f_{\arabic*}:x\mapsto{}$}, labelsep = 0pt]
  \item $x$
  \item $x^2$
  \item $x^3$
  \item $x^4$
  \item $x^5$
  \item $x^6$
  \item $x^7$
  \item $x^8$
  \item $x^9$
  \item $x^{10}$
\end{enumerate}

\end{document}

With the shortlabelsoption of enumitem, and two different alignments:

\documentclass{article}

\usepackage[shortlabels]{enumitem}
\usepackage[showframe]{geometry}
\usepackage{calc} 

\begin{document}

\begin{enumerate}[$f_\bgroup1\egroup\colon x\mapsto$, labelwidth=\widthof{$f_{99}\colon x\mapsto$}, leftmargin=!, labelsep =0.28em]
\item $x$
\item $x^2$
\item $x^3$
\item $x^4$
\item $x^5$
\item $x^6$
\item $x^7$
\item $x^8$
\item $x^9$
\item $x^{10}$
\end{enumerate}
\vspace{1cm}
\begin{enumerate}[$f_\bgroup1\egroup\colon x\mapsto$, wide=0pt, leftmargin=!, labelsep =0.28em]
\item $x$
\item $x^2$
\item $x^3$
\item $x^4$
\item $x^5$
\item $x^6$
\item $x^7$
\item $x^8$
\item $x^9$
\item $x^{10}$
\end{enumerate}

\end{document}