Functions Questions

The function $f(x)=\cos x-2px$ is monotonically decreasing for:

If $f:\mathbb{R}\to \mathbb{R}$ is a function such that:
$$f(x)=\{\begin{array}{ll}1,& \text{if }x>0\\ 0,& \text{if }x=0\\ -1,& \text{if }x<0\end{array}$$
then $f(x)$ is:

The maximum value of the function $f(x)=(x-1)(x-2)(x-3)$ is:

A function $f:\mathbb{R}\to \mathbb{R}$ satisfies the equation $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb{R}$; $f(x)\ne 0$. Suppose that the function $f(x)$ is differentiable at $x=0$ and $f^{\prime }(0)=2$. If $f^{\prime }(x)=\lambda f(x)$, then the value of $\lambda$ is:

The critical points of the function $f(x)=[x+1]-[x]+\frac{1}{x^2-1}$ are:

If $f(x)$ is an increasing function and $g(x)$ is a decreasing function, then which of the following is correct for the real values $p$ and $q$? Given that for these values of $x$, both functions are defined.

Let $S=\{(x,y):x^2+y^2=1,-1\le x\in \mathbb{R}\le 1\text{ and }-1\le y\in \mathbb{R}\le 1\}$. Which one of the following is correct?

Let $A=\{a,b,c,d\}$ and $B=\{1,2,3,4,5,6\}$. Then the number of one-to-one functions from $A$ to $B$ is:

Let $f$ and $g$ be differentiable functions on $\mathbb{R}$ such that $f\circ g$ is the identity function. If for some $a,b\in \mathbb{R}$, $g^{\prime }(a)=5$ and $g(a)=b$, then $f^{\prime }(b)$ is equal to:

If $f(x)=|x+1|+|x+10|$, then find the minimum value of $f(x)$.

If $f(x)=\{\begin{array}{ll}0,& x=0\\ x-3,& x>0\end{array}$, then the function $f(x)$ is:

Let $A=\{x\in \mathbb{R}:x\ge 0\}$. A function $f:A\to {\mathbb{R}}^{+}$ is defined by $f(x)=x^2$. Which one of the following is correct?

$f(x)=x+\sqrt{x^2}$ is a function from $\mathbb{R}\to \mathbb{R}$. Then $f(x)$ is:

What is the domain of the function $f(x)=\cos ({\sin }^{-1}(x+1))$?

If a function is defined as $f:\mathbb{N}\to \mathbb{N},f(x)=\frac{x-1}{2}$, then which of the following is true regarding this function?

Match List I with List II regarding the domain of functions:List I (Function)List II (Domain)A. $f(x)=\frac{1}{\sqrt{x-2}}+\frac{1}{\sqrt{3-x}}$ I. $[2,3]$ B. $f(x)=\sqrt{\frac{x-2}{3-x}}$ II. $(2,3]$ C. $f(x)=\sqrt{\frac{3-x}{x-2}}$ III. $[2,3)$ D. $f(x)=\sqrt{(x-2)(3-x)}$ IV. $(2,3)$ Choose the correct answer:

If $2f(x)+f\left(\frac{1}{x}\right)=\log x$, for all $x>0$, then $f(e^x)$ is: