Question

If the function $f:R\to R$, given by $f\left(x\right)=1+\alpha x$, where $\alpha \ne 0$, is the inverse of itself, then $\alpha$ =

• A. $-2$
• B. $-1$
• C. $3$
• D. $2$

Answer 1

The correct option is B $-1$

We have

$f\left(x\right)=1+\alpha x$

And given that $\frac{f\left(x\right)-1}{\alpha }=x$

So $f^{-1}\left(x\right)=\frac{x-1}{\alpha }$

But According to question
$f\left(x\right)=f^{-1}\left(x\right)$

$1+\alpha x=\frac{x}{\alpha }-\frac{1}{\alpha }$ So we have

$1=\frac{-1}{\alpha }$

$\Rightarrow \alpha =-1$