Jtdcftul

      documentclass[12pt,a4paper]{report}

  usepackage{mathtools, nccmath}

    usepackage{amssymb, amsthm, mathrsfs}



   begin{document}

    begin{align*}

   hcolon S(0,1)& rightarrow S^A(0,1)\

    x&mapsto frac{x}{|x|_A},

   end{align*}

   and

   begin{align*}

  gcolon S^A(0,1)& rightarrow S(0,1)\

 y&mapsto frac{y}{|y|},

end{align*}

   end{document}

I want to make the two equations in the same line

Use aligned:

documentclass[12pt,a4paper]{report}

usepackage{mathtools}



begin{document}



begin{equation*}

begin{aligned}[t]

  hcolon S(0,1) &rightarrow S^A(0,1)\

  x              &mapsto frac{x}{|x|_A},

end{aligned}

quadtext{and}quad

begin{aligned}[t]

  gcolon S^A(0,1) &rightarrow S(0,1)\

  y                &mapsto frac{y}{|y|},

end{aligned}

end{equation*}



end{document}

I used to be fond of this function notation, but then I started to dislike it: it's heavy, uses up space and doesn't really do anything good. On the blackboard it can be useful, not in papers.

Using aligned you can do the job in inline math mode. I suppose you would like the result centered, so, I placed the code in a center environment and finally, I added some hspace commands for better spacing:

 documentclass[12pt,a4paper]{report}

  usepackage{mathtools, nccmath}

    usepackage{amssymb, amsthm, mathrsfs}



   begin{document}

    begin{center}

    (begin{aligned}

   hcolon S(0,1)& rightarrow S^A(0,1)\

    x&mapsto frac{x}{|x|_A},

   end{aligned}

   hspace{20pt}text{ and }hspace{20pt}

   begin{aligned}

  gcolon S^A(0,1)& rightarrow S(0,1)\

 y&mapsto frac{y}{|y|},

end{aligned})

end{center}

   end{document}

Something like one of these?

documentclass[12pt,a4paper]{report}

usepackage{mathtools, nccmath}

usepackage{amssymb, amsthm, mathrsfs}



begin{document}



[ begin{aligned}

hcolon S(0,1)& longrightarrow S^A(0,1)\

x&longmapsto frac{x}{|x|_A},

end{aligned}

qquadtext{ and }qquad

begin{aligned}

gcolon S^A(0,1)& longrightarrow S(0,1)\

y&longmapsto frac{y}{|y|},

end{aligned} ]bigskip



[ begin{rcases} begin{aligned}

hcolon S(0,1)& longrightarrow S^A(0,1)\

x&longmapsto frac{x}{|x|_A},

end{aligned}end{rcases}

quadtext{ and }quad

begin{dcases} begin{aligned}

gcolon S^A(0,1)& longrightarrow S(0,1)\

y&longmapsto frac{y}{|y|},

end{aligned}end{dcases} ]



end{document}