Let $\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}$ and $\mathrm{g}: \mathbf{R} \rightarrow \mathbf{

Question

Let $\mathrm{f}: \mathbf{R} ightarrow \mathbf{R}$ and $\mathrm{g}: \mathbf{R} ightarrow \mathbf{R}$ be defined by $f(x)=\left\{\begin{array}{c}	ext { 0if } x 	ext { is rational } \ x 	ext { if } x 	ext { is irrational }\end{array}ight.$ \and $g(x)=\left\{\begin{array}{l}x 	ext { if } x 	ext { is rational } \ 0 	ext { if } x 	ext { is irrational }\end{array}ight.$.then $f-g$ is

Professor PC Thomas Classes · Answer

If x is rational $(\mathrm{f}-\mathrm{g})(\mathrm{x})=\mathrm{f}(\mathrm{x})-\mathrm{g}(\mathrm{x})=0-\mathrm{x}=-\mathrm{x}$ If x is irrational $ \ (f-g)(x)=f(x)-g(x)=x-0=x $ Clearly $\mathrm{f}-\mathrm{g}$ is one - one and onto