Question

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

• A. one-one onto
• B. many-one onto
• C. one-one but not onto
• D. none of these

Answer 1

Answer: (C)

one-one but not onto

$f\left(x\right)=x^2+x+1$
For $x\in N$
$f\left(x\right)\ge 3$
That is, its range of values includes all the natural numbers greater than and equal to 3.
However $f\left(x_1\right)=f\left(x_2\right)$
$\to x_1=x_2$
Hence $f\left(x\right)$ is a one-one function but not an onto function.