Question

Which of the following functions have their range equal to $R$(the set of real numbers)

• A. $x\sin x$
• B. $\frac{\left[x\right]}{\tan 2x}\cdot x\in (-\frac{\pi }{4}\cdot \frac{\pi }{4})-\left\{0\right\},[\cdot ]$ denotes the greatest integer function
• C. $\frac{x}{\sin x}$
• D. $\left[x\right]+\sqrt{\left\{x\right\}},[\cdot ]$ and $\{\cdot \}$ respectively denote the greatest integer and fractional part functions.

Answer 1

Answer: (A), (D)

The correct options are
A $x\sin x$
D $\left[x\right]+\sqrt{\left\{x\right\}},[\cdot ]$ and $\{\cdot \}$ respectively denote the greatest integer and fractional part functions.

$x\sin x$ is a continuous function value of $\left|x\sin x\right|$ can be made as a large as we like for sufficiently large values of x.
Therefore, range of $x\sin x=R$

$\frac{\left[x\right]}{\tan 2x}=0$

For $x\in (0,\frac{\pi }{4})$ as the $\left[x\right]=0$.

For $x\in (-\frac{\pi }{4},0),\frac{\left[x\right]}{\tan 2x}>0$.

Therefore, values of $\frac{\left[x\right]}{\tan 2x}$ are never negative.

Thus, range of $\frac{\left[x\right]}{\tan 2x}\ne R$.

$\left|\frac{x}{\sin x}\right|>1$, whenever defined.

Thus, range of $\frac{x}{\sin x}$ is not R.

$\left|x\right|+\sqrt{\left\{x\right\}}$ is a continuous function and

$\underset{x\to \infty }{lim}(\left[x\right]+\sqrt{\left\{x\right\}})=\infty ,\underset{x\to -\infty }{lim}\left(\right[x]-\sqrt{\left\{x\right\}})=-\infty$

Thus, range of $\left[x\right]+\left[x\right]=R$