First slide

Functions (XII)

Question

$Let f : N \to N$ be defined by $f (x) = x^{2} + x + 1, x \in N .$ Then f is

Easy

A

one-one onto
B

many-one onto
C

one-one but not onto
D

none of these

Solution

$Let x, y \in N such that f (x) = f (y) .$
$f (x) = f (y)$
or $x^{2} + x + 1 = y^{2} + y + 1$
or $(x - y) (x + y + 1) = 0$
i.e. $x = y o r x = (- y - 1) \notin N$
Therefore, f is one-one.
Also, f(x) does not take all positive integral values.so not onto
Hence, f is into.

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