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Question

Show that the function f:RR defines by f(x)=xx2+1,xR, is neither one-one nor onto.

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Solution

For x1,x2R such that x1x21
Consider f(x1)=f(x2)
x1x21+1=x2x22+1
x1x22+x1=x2x21+x2
x1x2(x2x1)(x2x1)=0
(x1x21)(x2x1)=0
x1x2=1 x1=1x2
Hence, f is not one-one as x1R there exists x2 such that f(x1)=f(x2)

Let, f(x)=y
xx2+1=y
y(x2+1)=x
yx2x+1=0
x=1±1dy22y

Since, xR 14y20
(12y)(12y)0
12y12
So, Range(f)[12,12]R
Hence, f is not into.

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