How to change page number positions for with active \maketitle

Question

I am trying to change the position of the page numbers from the bottom center to the bottom right. I tried using the fancyhdr package which seemed to be the most recommended way to solve this issue. However, when I used it along with the \maketitle command the fancyhdr package has no effect on the first page. On the second page the positioning of the numbers does move to the bottom right, but I do not like how it changes the format for the header. Below is a simplified version of my code. Any recommendations?

\documentclass[12pt]{article}
\usepackage{fullpage}
\usepackage{fancyhdr}

\title{Some Random Title}
\author{Author Name}
\date{}

\begin{document}
\maketitle
\pagestyle{fancy}
\fancyhf{}
\rfoot{Page \thepage}

Some random words.    
\newpage
Some random words on second page.

\end{document}

Answer 1

The page where \maketitle has effect uses the plain page style. Redefine it.

\documentclass[12pt]{article}
\usepackage{fullpage}
\usepackage{fancyhdr}

\pagestyle{fancy}
\fancyhf{}
\fancyfoot[R]{Page \thepage}

% redefine the plain page style
\fancypagestyle{plain}{%
  \renewcommand{\headrulewidth}{0pt}% no rule above!
  \fancyhf{}\fancyfoot[R]{Page \thepage}%
}

\title{Some Random Title}
\author{Author Name}
\date{}

\begin{document}
\maketitle

Some random words.

\end{document}