Consider a function $f: R \to R$ such that $f(x + a) = \frac{1}{2} + \sqrt{f(x) - f^2(x)}$,where $a$ is a real constant. Then $f(x)$ must be

Let $S = \mathbb{N} \cup \{0\}$. Define a relation $R$ from $S$ to $\mathbb{R}$ by: $R = \{(x, y) : \log_e y = x \log_e \left(\frac{2}{5}\right), x \in S, y \in \mathbb{R}\}$. Then,the sum of all the elements in the range of $R$ is equal to

Let $f(x)=x^2+2x+2$,$g(x)=-x^2+2x-1$,and $a, b$ be the extreme values of $f(x)$ and $g(x)$ respectively. If $c$ is the extreme value of $\frac{f}{g}(x)$ (for $x \neq 1$),then $a+2b+5c+4=$

Let $f(x) = [x]^2 - [x+3] - 3, x \in \mathbb{R}$,where $[\bullet]$ is the greatest integer function. Then:

Let $f:(-1,1) \rightarrow \mathbb{R}$ be such that $f(\cos 4 \theta) = \frac{2}{2-\sec^2 \theta}$ for $\theta \in \left(0, \frac{\pi}{4}\right) \cup \left(\frac{\pi}{4}, \frac{\pi}{2}\right)$. Then the value$(s)$ of $f\left(\frac{1}{3}\right)$ is (are):

Let $f(x) = Ax^3 - Bx - \tan x \cdot \text{sgn}(x)$ be an even function $\forall x \in R - \left\{ (2n + 1)\frac{\pi}{2}, n \in I \right\}$,where $A = \sin^2 \alpha - \sin \alpha + \frac{1}{4}$ and $B = \tan^2 \alpha + \frac{2}{\sqrt{3}} \tan \alpha + \frac{1}{3}$. Then the number of values of $\alpha$ in $\left[ -\frac{3\pi}{2}, 2\pi \right]$ is (where $\text{sgn}(x)$ denotes the signum function of $x$).