documentclass{article}
usepackage{tikz}
usepackage{pgfplots} % loads tikz
pgfplotsset{compat=1.15}
usetikzlibrary{intersections,fillbetween}
begin{document}
begin{tikzpicture}
    path[name path = line1, draw] (-1,0) to[out=90,in=90] (1,1) to[out=-90,in=0]  (0,-1);
    path[name path = line2, draw] (-1,1) -- (1,-1);
    draw[very thick, intersection segments={of=line1 and line2,sequence=L2}];
end{tikzpicture}

begin{tikzpicture}
    path[name path = line1, draw] (-1,0) to[out=90,in=90] (1,1) to[out=-90,in=0]  (0,-1);
    path[name path = line2, draw] (-1,1) -- (1,-1);
    draw[very thick, intersection segments={of=line1 and line2,sequence=L1 R2}];
end{tikzpicture}

% The following does not work.
%
% begin{tikzpicture}
%     path[name path = line1, draw] (-1,0) to[out=90,in=90] (1,1) to[out=-90,in=0]  (0,-1);
%     path[name path = line2, draw] (-1,1) -- (1,-1);
%     draw[very thick, intersection segments={of=line1 and
%   line2,sequence=L2[reverse] R2}];
% end{tikzpicture}

end{document}