Relation and Function — Class 12 Mathematics PYQs

Let $R$ be a relation on the open interval $(-1,1)$ and is given by
$R=\{(x,y):|x+y|<2\}$. Then which one of the following is correct?

Let $A=\{1,2,3,4,5\}$ and $B=\{6,7\}$. What is the number of onto functions from $A$ to $B$?

If $f(x)=ax-b$ and $g(x)=cx+d$ are such that $f(g(x))=g(f(x))$, then which one of the following holds?

Let $f(x)f(y)=f(xy)$ for all real $x,y$. If $f(2)=4$, then what is the value of $f(1/2)$?

Direction : Consider the following for the two (02) items that follow :

Let $f\circ g(x)={\cos }^2\sqrt{x}$ and $g\circ f(x)=|\cos x|$.

Which one of the following is $f(x)$?

Direction : Consider the following for the two (02) items that follow :

Let $f\circ g(x)={\cos }^2\sqrt{x}$ and $g\circ f(x)=|\cos x|$.

Which one of the following is $g(x)$?

Consider the following for the next two (02) items that follow:

Let $f(x)$ and $g(x)$ be two functions such that $g(x)=x-\frac{1}{x}$ and $f\circ g(x)=x^3-\frac{1}{x^3}$.

What is $g[f(x)-3x]$ equal to?

Let $A=\{1,2,3,\dots ,20\}$. Define a relation $R$ from $A$ to $A$ by $R=\{(x,y):4x-3y=1\}$, where $x,y\in A$. Which of the following statements is/are correct ?

1. The domain of $R$ is $\{1,4,7,10,13,16\}$.
2. The range of $R$ is $\{1,5,9,13,17\}$.
3. The range of $R$ is equal to codomain of $R$.

Select the correct answer using the code given below :

A mapping $f:A\to B$ defined as $f(x)=\frac{2x+3}{3x+5},x\in A$. If $f$ is to be onto, then what are $A$ and $B$ equal to?

Consider the following statements :

1. The relation $f$ defined by
   $f(x)=\{\begin{array}{ll}{x}^3,& 0\le x\le 2\\ 4x,& 2\le x\le 8\end{array}$
   is a function.

2. The relation $g$ defined by
   $g(x)=\{\begin{array}{ll}{x}^2,& 0\le x\le 4\\ 3x,& 4\le x\le 8\end{array}$
   is a function.

Which of the statements given above is/are correct ?