Question

For a real number $x$, let $\left[x\right]$ denote the greatest integer less than or equal to $x$. Let $f:R\to R$ be defined as $f\left(x\right)=2x+\left[x\right]+\sin x.\cos x$ then $f$ is

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

Answer 1

Answer: (C)

both one-one and onto

$f\left(x\right)=2x+\left[x\right]+\sin x\cos x$
Range of $f\left(x\right)$ will be
$\begin{array}{c}\underset{x\to -\infty }{lim}f\left(x\right)=-\infty \\ \underset{x\to \infty }{lim}f\left(x\right)=\infty \end{array}$
$\therefore$ Range$=\left(-\infty ,\infty \right)$
Range =R=Co-domain
$\therefore$ Function is onto
Now,
$\begin{array}{c}f\left(x_1\right)=2x_1+\left[x_1\right]+\sin x_1\cos x_1\\ f\left(x_2\right)=2x_2+\left[x_2\right]+\sin x_2\cos x_2\\ f\left(x_1\right)=f\left(x_2\right)\\ \Rightarrow 2x_1+\left[x_1\right]+\sin x_1\cos x_1=2x_2+\left[x_2\right]+\sin x_2\cos x_2\\ \Rightarrow x_1=x_2\end{array}$
So, f(x) is also one-one function