$($a$)$ Determine whether the mapping $f$:$QQ$ defined by $f(\frac{m}{n})=\frac{m^2}{n},$AAm,ninZ,$n0$ is well defined function. Justify your answer. $($b$)$ Let $f$:$(\frac{R}{\frac{?}{?}}\{1\})(\frac{R}{\frac{?}{?}}\{1\})$ be a function defined by $f(x)=\frac{x}{x-1},$AAxin$\frac{R}{\frac{?}{?}}\{1\}.$ Find $f^{-1}$ and verify that it is indeed an inverse function for the function $f.($c$)$ Determine whether or not the functions $f$:$RR$ defined as follows $f(x)=\frac{x}{x^2+1},$ is one$-$to$-$one.