Question

Let the function $f:R\to R$ be defined by $f\left(x\right)=2x+\sin x(x\in R).$ Then $f$ is

• A. one-to-one and onto
• B. one-to-one but not onto
• C. onto but not one-to-one
• D. neither one-to-one nor onto

Answer 1

The correct option is A one-to-one and onto

$f\left(x\right)=2x+sinx$

$f\left(x_1\right)=f\left(x_2\right)$ implies

$2x_1+sin\left(x_1\right)=2x_2+sin\left(x_2\right)$

Therefore

$2x_1=2x_2$

$x_1=x_2$ ...(i)

$sin{x}_1=sin{x}_2$

$x_1=n\pi +(-1{)}^n(x_2)$ ...(ii)

From i and ii, $x_1=x_2$

And all the elements of the co-domain are used.
Hence it is one one and onto.