Question

If the function f(x) = 1 + kx , x is not equal to 0, is the inverse of itself, then the value of k is

Answer 1

$\mathrm{Given}f\mathrm{is}\mathrm{the}\mathrm{inverse}\mathrm{of}\mathrm{itself}.\Rightarrow {\mathrm{fof}}^{-1}\left(x\right)=x\Rightarrow 1+{\mathrm{kf}}^{-1}\left(x\right)=x\Rightarrow f^{-1}\left(x\right)=\frac{x-1}{k}\mathrm{Since}f\mathrm{is}\mathrm{the}\mathrm{inverse}\mathrm{of}\mathrm{itself},f\left(x\right)=f^{-1}\left(x\right)\mathrm{for}\mathrm{all}x\in R\Rightarrow 1+\mathrm{kx}=\frac{x-1}{k}\mathrm{for}\mathrm{all}x\in R\Rightarrow \mathrm{kx}+1=\left(\frac{1}{k}\right)x+\left(-\frac{1}{k}\right)\mathrm{for}\mathrm{all}x\in R\mathrm{Comparing}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get}k=\frac{1}{k}\mathrm{and}1=-\frac{1}{k}\Rightarrow k^2=1\mathrm{and}k=-1\Rightarrow k=-1$