The function $f(x) = \sec \left[ \log \left( x + \sqrt{1 + x^2} \right) \right]$ is . . . . . . function.

Let $f(x) = x^{12} - x^9 + x^4 - x + 1$. Which of the following is true?

Let $c, k \in R$. If $f(x)=(c+1) x^{2}+(1-c^{2}) x+2 k$ and $f(x+y)=f(x)+f(y)-x y$,for all $x, y \in R$,then the value of $|2(f(1)+f(2)+f(3)+\ldots+f(20))|$ is equal to

Let $N$ be the set of positive integers. For all $n \in N$,let $f_n = (n+1)^{1/3} - n^{1/3}$ and $A = \{n \in N : f_{n+1} < \frac{1}{3(n+1)^{2/3}} < f_n\}$. Then,

Let $f, g: R \rightarrow R$ be defined,respectively,by $f(x) = x + 1$ and $g(x) = 2x - 3$. Find $f+g$,$f-g$,and $\frac{f}{g}$.

If $X$ and $Y$ are two non-empty sets where $f: X \to Y$ is a function defined such that $f(C) = \{f(x) : x \in C\}$ for $C \subseteq X$ and $f^{-1}(D) = \{x : f(x) \in D\}$ for $D \subseteq Y$,then for any $A \subseteq X$ and $B \subseteq Y$,which of the following is true?