Using Monotonicity to Find the Range of a Function

Q. The range of the function f(x) = sin [x], $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$ where [x] denotes the greatest integer, is

1. {0, $\pm$ sin 1}
2. {0, - sin 1}
3. {0}
4. {0, -1}

Q. $\text{The range of the function}f\left(x\right)=\frac{x+3}{|x+3|},x\ne -3is$

1. {-1, 1}

2. {3, -3}

3. R-{-3}

4. all positive integers

Q. If $2f\left(sinx\right)+f\left(cosx\right)=x\forall xϵ$ R then range of f(x) is

1. $[\frac{-\pi }{3},\frac{\pi }{3}]$
2. $[\frac{-2\pi }{3},\frac{\pi }{3}]$
3. $[\frac{-2\pi }{3},\frac{\pi }{6}]$
4. $[\frac{-\pi }{6},\frac{\pi }{6}]$

Q. If $f\left(x\right)=ln\left(\frac{x^2+e}{x^2+1}\right)$ , then range of f(x) is

1. $(0,1)$
2. $(0,1]$
3. $[0,1)$
4. $\{0,1\}$