Question

IF f:R to R be defined as follows:
F(x)= {1, if x belongs to Q}
F(x)={-1,if x does not belongs to Q}
find
1)f(1/2),f(pi), f(root 2)
2)range of f
3)pre images of 1 and -1

Answer 1

Dear student

$$\begin{gathered} f\left(x\right)=\left\{\begin{array}{ll}1& ,x\in Q\\ -1& ,x\notin Q\end{array}\right. \\ 1)f\left(\frac{1}{2}\right)=1as\frac{1}{2}\in Q \\ f\left(\mathrm{\pi }\right)=-1as\mathrm{\pi }\notin \mathrm{Q} \\ \mathrm{f}\left(\sqrt{2}\right)=-1\mathrm{as}\sqrt{2}\notin \mathrm{Q} \\ 2)\mathrm{Range}\mathrm{of}\mathrm{f}\mathrm{is}\mathrm{x}\in [-1,1] \\ 3)\mathrm{pre}-\mathrm{image}\mathrm{of}1\mathrm{is}\frac{1}{2}\mathrm{according}\mathrm{to}\mathrm{part}(1) \\ \mathrm{pre}-\mathrm{image}\mathrm{of}-1\mathrm{is}\mathrm{\pi }\mathrm{and}\sqrt{2}\mathrm{according}\mathrm{to}\mathrm{part}(1) \\ \mathrm{In}\mathrm{general}\mathrm{pre}-\mathrm{image}\mathrm{of}1\mathrm{is}\mathrm{x}\in \mathrm{Q} \\ \mathrm{and}\mathrm{pre}-\mathrm{image}\mathrm{of}-1\mathrm{is}\mathrm{x}\notin \mathrm{Q} \end{gathered}$$
Regards