If $A=\{-1,2,3\}$ and $B=\{1,3\}$, then determine

(i) $A \times B$

(ii) $B \times A$

(iii) $B \times B$

(iv) $A \times A$

If $P=\{x: x<3, x \in N\}, \quad Q=\{x: x \leq 2, x \in W\}, \quad$ then find $(P \cup Q) \times(P \cap Q)$, where $W$ is the set of whole numbers.

If $A=\{x: x \in W, x< 2\}, B=\{x: x \in N, 1< x< 5\}$ and $C=\{3,5\}$, then find

(i) $A \times(B \cap C)$

(ii) $A \times(B \cup C)$

In each of the following cases, find $a$ and $b$.

(i) $(2 a+b, a-b)=(8,3)$

(ii) $\left(\frac{a}{4}, a-2 b\right)=(0,6+b)$

$A=\{1,2,3,4,5\}, S=\{(x, y): x \in A, y \in A\}$, then find the ordered which satisfy the conditions given below.

(i) $x+y=5$

(ii) $x+y<5$

(iii) $x+y>8$

If $R=\left\{(x, y): x, y \in W, x^2+y^2=25\right\}$, then find the domain and range of $R$.

If $R_1=\{(x, y) \mid y=2 x+7$, where $x \in R$ and $-5 \leq x \leq 5\}$ is a relation. Then, find the domain and range of $R_1$.

If $R_2=\{x, y) \mid x$ and $y$ are integers and $\left.x^2+y^2=64\right\}$ is a relation, then find the value of $R_2$.

If $R_3=\{(x,|x|) \mid x$ is a real number $\}$ is a relation, then find domain and range of $R_3$.

Is the given relation a function? Give reason for your answer.

(i) $h=\{(4,6),(3,9),(-11,6),(3,11)\}$

(ii) $f=\{(x, x) \mid x$ is a real number $\}$

(iii) $g=\left\{\left(x, \frac{1}{x}\right) x\right.$ is a positive integer $\}$

(iv) $s=\left\{\left(x, x^2\right) \mid x\right.$ is a positive integer $\}$

(v) $t=\{(x, 3) \mid x$ is a real number $\}$

If $f$ and $g$ are real functions defined by $f(x)=x^2+7$ and $g(x)=3 x+5$. Then, find each of the following.

(i) $f(3)+g(-5)$

(ii) $f\left(\frac{1}{2}\right) \times g(14)$

(iii) $f(-2)+g(-1)$

(iv) $f(t)-f(-2)$

(v) $\frac{f(t)-f(5)}{t-5}$, if $t \neq 5$

Let $f$ and $g$ be real functions defined by $f(x)=2 x+1$ and $g(x)=4 x-7$.

(i) For what real numbers $x, f(x)=g(x)$ ?

(ii) For what real numbers $x, f(x)< g(x)$ ?

If $f$ and $g$ are two real valued functions defined as $f(x)=2 x+1$ and $g(x)=x^2+1$, then find

(i) $f+g$

(ii) $f-g$

(iii) $f g$

(iv) $\frac{f}{g}$

Express the following functions as set of ordered pairs and determine their range.

$$f: x \rightarrow R, f(x)=x^3+1 \text {, where } x=\{-1,0,3,9,7\}$$

Find the values of $x$ for which the functions $f(x)=3 x^2-1$ and $g(x)=3+x$ are equal.

Is $g=\{(1,1),(2,3),(3,5),(4,7),\}$ a function, justify. If this is described by the relation, $g(x)=\alpha x+\beta$, then what values should be assigned to $\alpha$ and $\beta$ ?

Find the domain of each of the following functions given by

(i) $f(x)=\frac{1}{\sqrt{1-\cos x}}$

(ii) $f(x)=\frac{1}{\sqrt{x+|x|}}$

(iii) $f(x)=x|x|$

(iv) $f(x)=\frac{x^3-x+3}{x^2-1}$

(v) $f(x)=\frac{3 x}{28-x}$

Find the range of the following functions given by

(i) $f(x)=\frac{3}{2-x^2}$

(ii) $f(x)=1-|x-2|$

(iii) $f(x)=|x-3|$

(iv) $f(x)=1+3 \cos 2 x$

Redefine the function

$$f(x)=|x-2|+|2+x|,-3 \leq x \leq 3$$

If $f(x)=\frac{x-1}{x+1}$, then show that

(i) $f\left(\frac{1}{x}\right)=-f(x)$

(ii) $f\left(-\frac{1}{x}\right)=\frac{-1}{f(x)}$

If $f(x)=\sqrt{x}$ and $g(x)=x$ be two functions defined in the domain $R^{+} \cup\{0\}$, then find the value of

i) $(f+g)(x)$

(ii) $(f-g)(x)$

(iii) $(f g)(x)$

(iv) $\left(\frac{f}{g}\right)(x)$

Find the domain and range of the function $f(x)=\frac{1}{\sqrt{x-5}}$.

If $f(x)=y=\frac{a x-b}{c x-a}$, then prove that $f(y)=x$.

Let $f=\{(2,4),(5,6),(8,-1),(10,-3)\}$ and $g=\{(2,5),(7,1),(8,4),(10,13),(11,5)\}$ be two real functions. Then, match the following.

The domain of $f-g, f+g, f \cdot g, \frac{f}{g}$ is domain of $f \cap$ domain of $g$. Then, find their images.

Let $n(A)=m$ and $n(B)=n$. Then, the total number of non-empty relations that can be defined from $A$ to $B$ is

If $[x]^2-5[x]+6=0$, where $[\cdot]$ denote the greatest integer function, then

Range of $f(x)=\frac{1}{1-2 \cos x}$ is

Let $f(x)=\sqrt{1+x^2}$, then

Domain of $\sqrt{a^2-x^2}(a>0)$ is

If $f(x)=a x+b$, where $a$ and $b$ are integers, $f(-1)=-5$ and $f(3)=3$, then $a$ and $b$ are equal to

The domain of the function $f$ defined by $f(x)=\sqrt{4-x}+\frac{1}{\sqrt{x^2-1}}$ is equal to

The domain and range of the real function $f$ defined by $f(x)=\frac{4-x}{x-4}$ is given by

The domain and range of real function $f$ defined by $f(x)=\sqrt{x-1}$ is given by

The domain of the function $f$ given by $f(x)=\frac{x^2+2 x+1}{x^2-x-6}$.

The domain and range of the function $f$ given by $f(x)=2-|x-5|$ is

The domain for which the functions defined by $f(x)=3 x^2-1$ and $g(x)=3+x$ are equal to

Let $f$ and $g$ be two real functions given by

$$\text { and } \quad \begin{aligned} f & =\{(0,1),(2,0),(3,-4),(4,2),(5,1)\} \\ & =\{(1,0),(2,2),(3,-1),(4,4),(5,3)\} \end{aligned}$$

then the domain of $f \cdot g$ is given by........... .

The ordered pair $(5,2)$ belongs to the relation $$R=\{(x, y): y=x-5, x, y \in Z\}$$

If $P=\{1,2\}$, then $P \times P \times P=\{(1,1,1),(2,2,2),(1,2,2),(2,1,1)\}$

$\begin{aligned} & \text { If } A=\{1,2,3\}, B=\{3,4\} \text { and } C=\{4,5,6\} \text {, then }(A \times B) \cup(A \times C) \\ & =\{(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,3) \text {, } \\ & (3,4),(3,5),(3,6)\} .\end{aligned}$

If $$(x - 2,y + 5) = \left( { - 2,{1 \over 3}} \right)$$ are two equ8al ordered pairs, then $$x = 4,y = {{ - 14} \over 3}$$

If $A \times B=\{(a, x),(a, y),(b, x),(b, y)\}$, then $A=\{a, b\}$ and $B=\{x, y\}$