Bijective Function

Q. Let $f:[-\frac{\pi }{2},\frac{\pi }{2}]\to R$ be a continuous function such that $f\left(0\right)=1$ and $\underset{0}{\overset{\pi /3}{\int }}f\left(t\right)dt=0$

Then which of the following statements is(are) TRUE ?

1. The equation $f\left(x\right)-3\cos 3x=0$ has at least one solution in $(0,\frac{\pi }{3})$
2. The equation $f\left(x\right)-3\sin 3x=-\frac{6}{\pi }$ has at least one solution in $(0,\frac{\pi }{3})$
3. $\underset{x\to 0}{lim}\frac{x\underset{0}{\overset{x}{\int }}f\left(t\right)dt}{1-e^{x^2}}=-1$
4. $\underset{x\to 0}{lim}\frac{\sin x\underset{0}{\overset{x}{\int }}f\left(t\right)dt}{x^2}=-1$

Q. The number of solutions of the equation $x+2\tan x=\frac{\pi }{2}$ in the interval $[0,2\pi ]$ is :

1. $5$
2. $2$
3. $4$
4. $3$

Q.

Check the injectivity and surjectivity of the following functions:
$\left(i\right)f:N\to N$ given by $f\left(x\right)=x^2$

$\left(ii\right)f:Z\to Z$ given by $f\left(x\right)=x^2$

$\left(ii\right)f:Z\to Z$ given by $f\left(x\right)=x^2$

$\left(iv\right)f:N\to N$ given by $f\left(x\right)=x^3$

$\left(v\right)f:Z\to Z$ given by $f\left(x\right)=x^3$

Q.

$f:R\to R,f\left(x\right)=x|x|is$

1. one-one but not onto

2. onto but not one-one

3. Both one-one and onto

4. neither one-one nor onto

Q. The function $f:A\to B$ defined by $f\left(x\right)=-x^2+6x-8$ is a bijection if

1. $A=(-\infty ,3]\text{and}B=(-\infty ,1]$
2. $A=[3,\infty )\text{and}B=(-\infty ,1]$
3. $A=(-\infty ,-3]\text{and}B=(-\infty ,-1]$
4. $A=[-3,\infty )\text{and}B=[1,\infty )$

Q. The function $f:(-\infty ,-1]\to (0,e^5]$ defined by $f\left(x\right)=e^{x^3+3x+2}$ is

1. a one-one function
2. a many-one function
3. an into function
4. an onto function

Q.

The greatest value of $\mathrm{f}\left(\mathrm{x}\right)={(\mathrm{x}+1)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-{(\mathrm{x}-1)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ on $\left[0,1\right]$ is

1. $1$

2. $2$

3. $3$

4. $4$

Q. Find the domain and range of the following real functions:
$\left(i\right)f\left(x\right)=-|x|\left(ii\right)f\left(x\right)=\sqrt{9-x^2}$

Q. Let f : N → N be defined by State whether the function f is bijective. Justify your answer.

Q. In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) f : R → R defined by f ( x ) = 3 − 4 x (ii) f : R → R defined by f ( x ) = 1 + x 2

Q. Prove that the function $f:N\to N,$ defined by $f\left(x\right)=x^2+x+1$ is one-one but not onto. Find inverse of $f:N\to S$, where $S$ is range of $f.$

Q.

Is The Modulus of A Complex Number Always Positive?

Q.

If $f\left(x\right)={\left(x+1\right)}^2-1,x\ge -1$. Then,

Statement-I: The set $\left\{x:f\left(x\right)=f^{-1}\left(x\right)\right\}=\left\{0,-1\right\}$.

Statement-II: $f$ is a bijection.

1. Statement-I is correct and Statement-II is correct; Statement-II is a correct explanation for Statement-I.

2. Statement-I is correct and Statement-II is correct; Statement-II is not a correct explanation for Statement-I.

3. Statement-I is correct and Statement-II is incorrect;

4. Statement-I is incorrect and Statement-II is incorrect;

Q. Let $f\left(x\right)=\frac{(2^x+1{)}^2}{2^x}$ and $g\left(x\right)={\sin }^3{x}^3.$
If $fog$ and $gof$ are defined, then which of the following is/are correct

1. $fog$ is an even function
2. $fog$ is an odd function
3. $gof$ is an odd function
4. $gof$ is an even function

Q.

Let $f:X\to Y$ be an invertible function. Show that the inverse of $f^{-1}$ is f i.e., $(f^{-1}{)}^{-1}=f.$

Q. The number of bijection functions that can be defined from set $A$ to set $B$ is $24,$ then $n\left(A\right)+n\left(B\right)$ is

Q. Which of the following functions are bijective?

1. $f:R\to [0,\infty )$ defined as $f\left(x\right)=e^{\text{sgn}\left(x\right)}$
2. $f:R\to R$ defined by $f\left(x\right)=\left[x\right]+\cos \left(\pi \right[x\left]\right)$ where $[.]$ denotes the greatest integer function
3. $f:R\to R$ defined as $f\left(x\right)=min\{x+2,-2x+4\}$
4. $f:[3,4]\to [4,6]$ defined by $f\left(x\right)=|x-1|+|x-2|+|x+3|+|x-4|$

Q. Let the function $f:[0,\infty )\to R$ be defined by $f\left(x\right)=2x+\sin x.$ Then $f$ is

1. bijective function
2. injective but not surjective
3. surjective but not injective
4. neither injective nor surjective

Q.

If $f\left(x\right)$ is differentiable and $f\left(0\right)=a$, then $\underset{x\to 0}{\lim }\frac{2f\left(x\right)-3f\left(2x\right)+f\left(4x\right)}{x^2}=$

1. $3a$

2. $2a$

3. $5a$

4. $4a$

Q. If $f:R\to R$ be defined by $f\left(x\right)=e^x$ and $g:R\to R$ be defined by $g\left(x\right)=x^2.$ The mapping $g\circ f:R\to R$ be defined by $(g\circ f)\left(x\right)=g\left[f\right(x\left)\right]\forall x\in R,$ Then

1. $g\circ f$ is bijective but $f$ is not injective
2. $g\circ f$ is injective and $g$ is injective
3. $g\circ f$ is injective but $g$ is not bijective
4. $g\circ f$ is surjective and $g$ is surjective

Q. The number of constant functions possible from $R$ to $B$ where $B=\{2,4,6,8,...24\}$ are

1. $24$
2. $12$
3. $8$
4. $6$

Q.

Let A =R -{3}, B=R -{1}. If $f:A\to B$ be defined by $f\left(x\right)=\frac{x-2}{x-3},\forall x\in A$. Then, show that f is bijective.

Q.

What are transitive relations?

Q.

Let $E=\left\{1,2,3,4\right\}$ and $F=\left\{1,2\right\}$. Then the number of onto functions from $E$ to $F$is?

Q. Show that the function $f:R\to R$ defined by $f\left(x\right)=\frac{x}{x^2+1}$ $\forall x\in R$ is neither one-one nor onto.

Q. Let a function $f\left(x\right)=\{\begin{array}{cc}{a}^2-2a-3+5x^3,& x<0\\ -2x^2,& x\ge 0\end{array}$ has a local maximum at $x=0.$ Then set of values of $a$ is

1. $(-\infty ,-1]$
2. $[-1,3]$
3. $(-\infty ,3]$
4. $[3,\infty )$

Q.

Consider $f:R_{+}\to [-5,\infty )$ given by $f\left(x\right)=9x^2+6x-5$ show that f is ivnertible with $f^{-1}\left(y\right)=\left(\frac{\left(\sqrt{y+6}\right)-1}{3}\right)$

Q.

If $m_1,m_2,m_3$ and $m_4$ respectively denote the moduli of the complex numbers $1+4i,3+i,1-i$ and $2-3i$, then the correct one, among the following is:

1. $m_1<m_2<m_3<m_4$

2. $m_4<m_3<m_2<m_1$

3. $m_3<m_2<m_4<m_1$

4. $m_3<m_1<m_2<m_4$

Q. If $0<a<b,$ then which of the following is correct :

1. $\frac{(b-a)}{a}<\ln \frac{b}{a}<\frac{(b-a)}{b}$
2. $\frac{(b-a)}{b}<\ln \frac{b}{a}<\frac{(a-b)}{b}$
3. $\frac{(b-a)}{b}<\ln \frac{b}{a}<\frac{(b-a)}{a}$
4. $\frac{(b-a)}{a}<\ln \frac{b}{a}<\frac{(a-b)}{a}$

Q.

Describe Absolute Value Function