A real valued function $f: A \rightarrow B$ defined by $f(x)=\frac{4-x^2}{4+x^2} \forall x \in A$ is a bijection. If $-4 \in A$, then $A \cap B=$

Let $f(x)=x^2+2 b x+2 c^2$ and $g(x)=-x^2-2 c x+b^2 . x \in R$. If $b$ and $c$ are non-zero real numbers such that min $f(x)>\max g(x)$, then $\left|\frac{c}{b}\right|$ lies in the interval

If $f: R \rightarrow A$, defined by $f(x)=\cos x+\sqrt{3} \sin x-1$ is an onto function then $A=$

Let $g(x)=1+x-[x]$ and ${ }^{\prime}$

$ f(x)= \begin{cases}-1, & x<0 \\ 0, & x=0,[x] \text { denotes the greatest integer less } \\ 1, & x>0\end{cases} $

than or equal to $x$. Then for all $x, f(g(x))=$

1. Let [ $x$ ] represent the greatest integer less than or equal to $x,\{x\}=x-[x] \sqrt{2}=1.414$ and $\sqrt{3}=1.732$. If $f(x)=\left\{x+\left[\frac{x}{1+x^2}\right]\right\}$ is a real valued function, then $f(\sqrt{2})+f(-\sqrt{3})=$

If the range of the function $f(x)=-3 x-3$ is $\{3,-6,-9,-18\}$, then which one of the following is not in the domain of $f$ ?

If $f(x)=(x+1)^2-1, x \geq-1$, then $\left\{x \mid f(x)=f^{-1}(x)\right\}$ is

$ \text { Consider the following statements. } $

$ \begin{array}{cl} \hline \text { Statement I } & \begin{array}{l} \text { A function } f: A \rightarrow B \text { is said to be one-one if and } \\ \text { only if } f(x) \neq f(y) \Rightarrow x \neq y \end{array} \\ \hline \text { Statement II } & \begin{array}{l} \text { A relation } f: A \rightarrow B \text { is said to be a function if } x \neq y \\ \Rightarrow f(x) \neq f(y) \end{array} \\ \hline \end{array} $

Then, which one of the following is true?

The set of all real values of $x$ for which $f(x)=\sqrt{\frac{|x|-2}{|x|-3}}$ is a well defined function is

Let $f: N \rightarrow N$ be a function such that $f(x+y)=f(x)+f(y)+x y$ for every $x, y \in N$. If $f(\mathbb{l})=2$, then $\sum_{k=0}^{10} f(k)=$

If a real valued function $f:[-1,2] \rightarrow B$ defined by

$ f(x)= \begin{cases}1-x, & \text { when }-1 \leq x \leq 1 \\ x-1, & \text { when } 1 < x \leq 2\end{cases} $

is a surjection, then $B=$

The sum of the least positive integer and the greatest negative integer in the range of the function $f(x)=\frac{x^2-5 x+7}{x^2-5 x-7}$ is

The interval in which the curve represented by $f(x)=2 x+\log \left(\frac{x}{2+x}\right)$ is

The set of real values of $x$ such that $f(x)=\sqrt{\frac{[x]-1}{\left.[x]^2-[x]-6\right]}}$ is a real valued function is

If a function $f: Z \rightarrow Z$ is defined by $f(x)=x-(-1)^x$, then $f(x)$ is

Domain of the real valued function $f(x)=\log \left(x^2-1\right)+x \operatorname{coth}^{-1} x$ is

The domain and range of a real valued function $f(x)=\cos x-3$ are respectively

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are two functions defined by $f(x)=2 x-3$ and $g(x)=5 x^2-2$, then the least value of the function $(g \circ f)(x)$ is

Which of the following function are odd?

I. $f(x)=x\left(\frac{e^x-1}{e^x+1}\right)$

II. $f(x)=k^x+k^{-x}+\cos x$

III. $f(x)=\log \left(x+\sqrt{x^2+1}\right)$

Define the function, $f, g$ and $h$ from $R$ to $R$ such that $f(x)=x^2-1, g(x)=\sqrt{x^2+1}$ and $h(x)= \begin{cases}0, \text { if } & x \leq 0 \\ x, \text { if } & x \geq 0\end{cases}$ consider the following statements

(i) fog is invertible

(ii) $h$ is an identify function

(iii) $f \circ g$ is not invertible

(iv) $(h \circ f \circ g) x=x^2$

Then, which one of the following is true ?

Let $a > 1$ and $0 < \mathrm{b} < 1$. If $f: R \rightarrow[0,1]$ is defined by $f(x)=\left\{\begin{array}{ll}a^x, & -\infty < x < 0 \\ b^x, & 0 \leq x < \infty\end{array}\right.$, then $f(x)$ is

$f(x)=\log \left(\left(\frac{2 x^2-3}{x}\right)+\sqrt{\frac{4 x^4-11 x^2+9}{|x|}}\right) \text { is }$

Let $f: R-\left\{\frac{-1}{2}\right\} \rightarrow R$ be defined by $f(x)=\frac{x-2}{2 x+1}$. If $\alpha$ and $\beta$ satisfy the equation $f(f(x))=-x$, then $4\left(\alpha^2+\beta^2\right)=$

The domain of the real valued function $f(x)=\sin \left(\log \left(\frac{\sqrt{4-x^2}}{1-x}\right)\right.$ is

The range of the real valued function $f(x)=\sqrt{\frac{x^2+2 x+8}{x^2+2 x+4}}$ is

If $f(x)=\sqrt{2-x^2}$ and $g(x)=\log (1-x)$ are two real valued functions, then the domain of the function $(f+g)(x)$ is

Let $f(x)=(x+2)^2-2, x \geq-2$. Then, $f^{-1}(x)$ is equal to

If $f$ is the greatest integers function defined on $R$ as $f(x)=[x]$ and $g$ is the modulus function defined on $R$ as $g(x)=|x|$, then the value of $(g \circ f)\left(\frac{-5}{3}\right)$ is

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are two functions defined by $f(x)=a x+b(a \neq 0), \forall x \in R$ and $g(x)=c x^3+d(c \neq 0), \forall x \in R$, then $(f \circ g)^{-1}(x)$ is equal to

If $f(10-x)=3 x^2+4 x-5$ and $f(x)=p x^2+q x+r$, then $p+q+r$ is equal to

$f(x)=\sin x+\cos x \cdot g(x)=x^2-1$, then $g(f(x))$ is invertible if

If $f: z \rightarrow z$ is defined by $f(x)=x^9-11 x^8-2 x^7+22 x^6+x^4 -12 x^3+11 x^2+x-3, \forall x \in z$, then $f(11)$ is equal to

Let $f(x)=x^3$ and $g(x)=3^x$, then the quadratic equation whose roots are solutions of the equation $(f \circ g)(x)=(g \circ f)(x)$ (for $x \neq 0$) is

The real valued function $f(x)=\frac{x}{e^x-1}+\frac{x}{2}+1$ defined on $R /\{0\}$ is

The domain of the function $f(x)=\frac{1}{[x]-1}$, where $[x]$ is greatest integer function of $x$ is

Let $f: R \rightarrow R$ be a function defined by $f(x)=\frac{4^x}{4^x+2}$, what is the value of $f\left(\frac{1}{4}\right)+2 f\left(\frac{1}{2}\right)+f\left(\frac{3}{4}\right)$ is equal to

Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=2 x+1$ and $g(x)=x^2-2$ determine $(g \circ f)(x)$ is equal to

Given, the function $f(x)=\frac{a^x+a^{-x}}{2},(a>2)$, then $f(x+y)+f(x-y)$ is equal to

If $f$ is a function defined on $(0,1)$ by $f(x)=\min \{x-[x],-x-[x]\}$, then $(f \circ f o f o f)(x)$ is equal to $\rightarrow([\cdot]$ greatest integer function)

If ${({x^2} + 5x + 5)^{x + 5}} = 1$, then the number of integers satisfying this equation is

If $\frac{x^4}{(x-1)(x-2)}=f(x)+\frac{A}{x-1}+\frac{B}{x-2}$, then

Which statement among the following is true?

(i) the function $f(x)=x|x|$ is strictly increasing on $R-\{0\}$.

(ii) the function $f(x)=\log _{(1 / 4)} x$ is strictly increasing on $(0, \infty)$.

(iii) a one-one function is always an increasing function.

(iv) $f(x)=x^{1 / 3}$ is strictly decreasing on $R$