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Let f(x)=(alphax)/((x+1)),x!=-1. for wha...

Let `f(x)=(alphax)/((x+1)),x!=-1.` for what value of `alpha` is `f(f(x))=x ?` (a)`sqrt(2)` (b) `-sqrt(2)` (c) `1` (d) `-1`

A

`sqrt2`

B

`-sqrt2`

C

1

D

`-1`

Text Solution

Verified by Experts

The correct Answer is:
D
We have
`f(f(x))=x " " "for all "x ne -1`
`Rightarrow f((alpha x)/(x+1))=x "for all "xne -1`
`Rightarrow (alpha((alpha x)/(x+1)))/((alpha x)/(x+1)+1)=x" for all"xne-1`
`Rightarrow (alpha^(2)x)/(alphax+x+1)=x"for all"x ne-1`
`Rightarrow alpha^(2)x=(alpha+1)x^(2)+x"for all "x ne-1`
`Rightarrow (alpha+1)x^(2)+(1-alpha^(2))x=0"for all "xne-1`
`Rightarrow alpha+1=0 and 1-alpha^(2)=0`
`Rightarrow alpha=-1`
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