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Question

Let f:NN defined by f(x)=x2+x+1, xN then f is

A
One-one onto
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B
Many-one onto
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C
One-one but not onto
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D
None of the above
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Solution

The correct option is D One-one but not onto
Let x,yN such that f(x)=f(y)
x2+x+1=y2+y+1
x2+x=y2+y
x2+x+xy=y2+y+xy
x(x+y+1)=y(y+x+1)
(xy)(x+y+1)=0
x=y or x=y1
But x=(y1)=(y+1) does not belong to N
x=y
f is one -one
Also, f is not onto

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