Let $f: R \to R$ be a function. Define $g: R \to R$ by $g(x) = |f(x)|$ for all $x$. Then $g$ is

Let $f(x) = |x|$ and $g(x) = |x| + a$,where $a > 0$. For $0 \leq x \leq b$,the set $\{(x, y) \mid g(x) \leq y \leq f(x)\}$ represents all the points in the interior of:

If $f(x) = x \left( \frac{1}{x-1} + \frac{1}{x} + \frac{1}{x+1} \right)$ for $x > 1$, then:

Consider the following statements:
$(i)$ $A$ relation is a special case of a function.
$(ii)$ $A$ function is a special case of a relation.
$(iii)$ Both relation and function are the same.

Let $N$ be the set of all natural numbers and $f: N \rightarrow N$ be such that $1990 < f(1990) < 2100$ and satisfies the equation $x-f(x)=19[\frac{x}{19}]-90[\frac{f(x)}{90}]$,where $[y]$ denotes the greatest integer less than or equal to $y$. Then the number of possible values of $f(1990)$ is

Let $f: R \rightarrow R$ be defined by $f(x)=2x+3$. If $\alpha$ and $\beta$ are the roots of the equation $f(x^2)-2f(\frac{x}{2})-1=0$,then $\alpha^2+\beta^2=$