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Question

Let f(x)=kxx+1, x1.If f(f(x))=x for all x

1, then the value of k is


A
k=1
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B
k=0
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C
k=1
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D
k=2
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Solution

The correct option is C k=1
f(x)=kxx+1, x1.f(f(x))=f(kxx+1)=k(kxx+1)kxx+1+1=k2xx+1kx+x+1x+1=k2xx(k+1)+1f(f(x))=xk2xx(k+1)+1=xk2x=x2(k+1) + xx2(k+1) + x(1k2)=0x2(k+1) + x(1k)(1+k)=0x(k+1) (x+1k]=0k+1=0k=1


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