Using Monotonicity to Find the Range of a Function

Q.

If $f\left(x\right)=\sin x-\frac{x}{2}$​ is increasing function, then

1. $0<x<\frac{\mathrm{\pi }}{3}$

2. $-\frac{\mathrm{\pi }}{3}<x<0$

3. $-\frac{\mathrm{\pi }}{3}<x<\frac{\mathrm{\pi }}{3}$

4. $x=\frac{\mathrm{\pi }}{2}$

Q. Let $f:(1,3)\to R$ be a function defined by $f\left(x\right)=\frac{x\left[x\right]}{x^2+1}$, where $\left[x\right]$ denotes the greatest integer $\le x$. Then the range of $f$ is :

1. $(\frac{2}{5},\frac{3}{5}]\cup (\frac{3}{4},\frac{4}{5})$
2. $(\frac{2}{5},\frac{4}{5}]$
3. $(\frac{3}{5},\frac{4}{5})$
4. $(\frac{2}{5},\frac{1}{2})\cup (\frac{3}{5},\frac{4}{5}]$

Q. The range of $si{n}^{-1}x-co{s}^{-1}x$ is

1. $[\frac{-3\pi }{2},\frac{\pi }{2}]$
2. $[\frac{-5\pi }{2},\frac{\pi }{3}]$
3. $[\frac{-3\pi }{2},\pi ]$
4. $[0,\frac{\pi }{2}]$

Q. The range of $f\left(x\right)={\sin }^{-1}\left(\frac{x^2+1}{x^2+2}\right)$ is

1. $[0,\frac{\pi }{2}]$
2. $(0,\frac{\pi }{6})$
3. $[\frac{\pi }{6},\frac{\pi }{2})$
4. None of these

Q.

For real $x$, let $f\left(x\right)=x^3+5x+1$, then

1. f is onto R but not one-one
   

2. f is one-one and onto R
   

3. f is neither one-one nor onto R
   

4. f is one-one but not onto R

Q. The domain of the function $f\left(x\right)={\left({\log }_3\frac{1}{|\cos x|}\right)}^{\frac{1}{2020}}$ is

1. $R-\{\frac{n\pi }{2},n\in Z\}$
2. $R-\{n\pi ,n\in Z\}$
3. $R$
4. $R-\left\{\right(2n+1)\frac{\pi }{2},n\in Z\}$

Q.

Let $f,g$ and $h$be real valued functions defined on the interval [0, 1] by$f\left(x\right)=e^{x^2}+e^{-x^2},g\left(x\right)=x{e}^{x^2}+e^{-x^2}$ and $h\left(x\right)=x^2{e}^{x^2}+e^{-x^2}$. If a, b and c denote, respectively, the absolute maximum of f, g and h on [0, 1], then

1. a = v and c $\ne$ b

2. a = c and a $\ne$ b

3. a $\ne$ b and c$\ne$ b

4. a = b = c