Using Monotonicity to Find the Range of a Function

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. $\{0,1\}$

Q. If $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x-\frac{1}{2}\sin 2x,$ then the range of $f\left(x\right)$ is

1. $[0,\frac{3}{2}]$
2. $[-\frac{1}{2},\frac{7}{2}]$
3. $[0,\frac{9}{8}]$
4. $[\frac{3}{4},\frac{7}{8}]$

Q. $f$ is a continuous function in $[a,b];$ $g$ is a continuous function in $[b,c]$. A function $h\left(x\right)$ is defined as
$h\left(x\right)=f\left(x\right)$ for $xϵ[a,b)$
$=g\left(x\right)$ for $xϵ(b,c]$.
If $f\left(b\right)=g\left(b\right)$, then

1. $h\left(x\right)$ has a removable discontinuity at $x=b$
2. $h\left(x\right)$ may or may not be continuous in $[a,c]$
3. $h\left(b^{+}\right)=g\left(b^{+}\right)=f\left(b^{+}\right)$
4. $h\left(b^{-}\right)=g\left(b^{+}\right)=f\left(b^{-}\right)$

Q. Let $f:R\to \left[0.\frac{\pi }{2}\right)$ be defined by $f\left(x\right)=ta{n}^{-1}(x^2+x+a)$. Then the set of values of a for which f is onto is

1. $[0,\infty )$
   
2. $[\frac{1}{4},\infty )$
   
3. $(-\infty ,\frac{1}{4}]$
   
4. $\left\{\frac{1}{4}\right\}$

Q. $f\left(x\right)=\left\{\frac{|x+5|-2}{x^2+10x+21}\right\},$where {.} denotes fractional part function then which of the following is true?

1. Number of integers in the domain of f(x) is 2.
2. Number of integers in the domain of f(x) is 3.
3. Range of f(x) is $(0,\frac{1}{2}]$
4. Number of integers in the domain of f(x) is infinite.

Q.

How do you find the value of $\sin \left(\frac{5\pi }{12}\right)$?