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Let f(x)=(alpha x)/(x+1), x ne -1. Then,...

Let `f(x)=(alpha x)/(x+1), x ne -1.` Then, for what value of `alpha " is " f[f(x)]=x` ?

A

`sqrt(2)`

B

`-sqrt(2)`

C

1

D

-1

Text Solution

Verified by Experts

The correct Answer is:
D
Given, `f(x)=(alphax)/(x+1)`
`f[f(x)]=f((alphax)/(x+1))=(alpha((alphax)/(x+1)))/((alphax)/(x+1)+1)`
`=((alpha^(2)x)/(x+1))/((alphax+(x+1))/(x+1))=(alpha^(2)x)/((alpha+1)x+1)=x` [given] ...(i)
`rArr alpha^(2)x=(alpha +1)x^(2)+x`
`rArr x[alpha^(2)-(alpha +1)x-1]=0`
`rArr x(alpha +1)(alpha -1-x)=0`
`rArr alpha -1=0 and alpha +1=0`
`rArr alpha = -1`
But `alpha =1` does nto satisfy the Eq. (i).
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