Onto Function

Q. Let E = {1, 2, 3, 4} and F = {1, 2}. Then, the number of onto functions from E to F is

1. 14
2. 16
3. 12
4. 8

Q. Let a function f defined from $R\to R$ as $f\left(x\right)=\{\begin{array}{cc}2m-x,& x\le 1\\ 4mx+1,& x>1\end{array}$
If function is onto on R, then range of m is

1. $[-1,\infty )$
2. $[-1,0)$
3. {-1}
4. $(0,\infty )$

Q. Let $f,g:R\to R$ be functions defined by $f\left(x\right)=\{\begin{array}{cc}\left[x\right],& x<0\\ |1-x|,& x\ge 0\end{array}$ and $g\left(x\right)=\{\begin{array}{cc}{e}^x-x,& x<0\\ (x-1{)}^2-1,x\ge 0& \end{array}$

where $\left[x\right]$ denote the greatest integer less than or equal to $x$. Then, the function $fog$ is discontinuous at exactly :

1. four points
2. one point
3. two points
4. three points

Q. Let $A=\{1,3,5,7\},$ $B=\{2,4,6,8\}$ and $f:A\to B.$ Then number of functions $f$ such that $f\left(i\right)\ne i+1,\forall i=1,3,5,7$ is

1. $64$
2. $24$
3. $256$
4. $81$

Q. Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of onto functions from A into B is

1. $nP2$
2. $2^n-1$
3. $2^n-2$
4. $\left(n-1\right)!$

Q. Let a function f defined from $R\to R$ as $f\left(x\right)=\{\begin{array}{cc}x+p^2,& x\le 2\\ 2px+5,& x>2\end{array}$.
If the function is surjective, then sum of all possible integral value of p in [-100, 100] is

Q.

Using identities, evaluate ${998}^2$

Q. If set $A$ has $3$ elements and set $B$ has $4$ elements, then number of injective functions that can be defined from $A$ to $B$ is

1. $144$
2. $12$
3. $24$
4. $64$

Q. the range of the function f(x) = sin cos (ln(x^2+1/x^2+e) is

Q. Let $A=\{1,2,3,....,10\}$ and $f:A\to A$ be defined as $f\left(k\right)=\{\begin{array}{cc}k+1& \text{if}k\text{is odd}\\ k& \text{if}k\text{is even}\end{array}.$ Then the number of possible functions $g:A\to A$ such that $gof=f$ is:

1. ${10}^5$
2. ${}^{10}{C}_5$
3. $5^5$
4. $5!$

Q. If $\text{sgn}\left(y\right)$ denotes the signum function of $y,$ then the range of $f\left(x\right)=\text{sgn}(x^2+4x+5)$ is

1. $[1,\infty )$
2. $\{-1,0,1\}$
3. $\left\{1\right\}$
4. $\{0,1\}$

Q. Let $f:R\to R$ be a differentiable function with $f\left(0\right)=0.$ If $y=f\left(x\right)$ satisfies the differential equation
$\frac{dy}{dx}=(2+5y)(5y-2)$
then the value of $\underset{x\to -\infty }{lim}f\left(x\right)$ is

Q.

Using identities, evaluate ${99}^2$

Q.

Let $f$ and $g$ be differentiable functions on $R$, such that $fog$ is the identity function. If for some $a,b\in R$, $g\left(a\right)=5$ and $g\left(a\right)=b$, then $f\left(b\right)$ is equal to

1. $\frac{2}{5}$

2. $5$

3. $1$

4. $\frac{1}{5}$

Q. Find the number of all onto functions from the set {1, 2, 3, … , n ) to itself.

Q. For any real valued function satisfying f(x)-sin(x)(f(x)-1)<=0 for all real x and f(0)=1, then find the range of f(x). (A)(-infinity, 1] (B)[-1, 1] (C)[1, infinity) (D)(-1, 1)

Q. If $f:R\to (-\infty ,a]$ defined by $f\left(x\right)=-x^2+6x+15$ is surjective, then the value of $a$ is

Q. Let $R_1$ and $R_2$ be two relations defined as follows:
$R_1=\left\{\right(a,b)\in R^2:a^2+b^2\in Q\}$ and
$R_2=\left\{\right(a,b)\in R^2:a^2+b^2\notin Q\}$, where $Q$ is the set of all rational numbers. Then :

1. $R_1$ is transitive but $R_2$ is not transitive.
2. $R_1$ and $R_2$ are both transitive.
3. $R_2$ is transitive but $R_1$ is not transitive.
4. Neither $R_1$ nor $R_2$ is transitive.

Q. Check the injectivity and surjectivity of the following functions: (i) f : N → N given by f ( x ) = x 2 (ii) f : Z → Z given by f ( x ) = x 2 (iii) f : R → R given by f ( x ) = x 2 (iv) f : N → N given by f ( x ) = x 3 (v) f : Z → Z given by f ( x ) = x 3

Q. Let $R\to R$ be function defined as
$f\left(x\right)=a\sin \left(\frac{\pi \left[x\right]}{2}\right)+[2-x],a\in R$, where $\left[t\right]$ is the greatest integer less than or equal to $t$. If $\underset{x\to -1}{lim}f\left(x\right)$ exists, then the value of $\underset{0}{\overset{4}{\int }}f\left(x\right)dx$ is equal to

Q. Let A = R − {3} and B = R − {1}. Consider the function f : A → B defined by . Is f one-one and onto? Justify your answer.

Q. Prove that the function $f$ given by $f\left(x\right)=x^2-x+1$ is neither strictly increasing nor strictly decreasing on $(-1,1)$.

Q. Which of the following functions are onto functions?

1. $f:R\to [-1,1]$ as $f\left(x\right)=sin\left(x\right)$
2. $f:R\to [-1,1]$ as $f\left(x\right)=cos\left(x\right)$
3. $f:R\to (0,\infty )$ as $f\left(x\right)=e^x$
4. $f:R\to (-\infty ,\infty )$ as $f\left(x\right)=x^3$

Q.

What type of function is the sine function in R ?

1. many one

2. one-one and onto

3. neither onto nor one-one

4. one-one

Q. There are exactly two distinct linear functions, which map [-1, 1] on to [0, 2], they are

1. x+1, x-1
2. -1+x, -1-x
3. x+1, 1-x
4. x+1, -x-1

Q. Let $f:R\to R$ be a function defined by $f\left(x\right)=x^3+x^2+3x+\sin x.$ Then $f$ is

1. injective and surjective
2. injective but not surjective
3. neither injective nor surjective
4. surjective but not injective

Q. Which among the following functions are invertible?

1. $f:Z\to Z$ defined by $f\left(x\right)=|x|$
2. $f:Z\to Z$ defined by $f\left(x\right)=x+2$
3. $f:Z\to Z$ defined by $f\left(x\right)=x^3$
4. $f:Z\to Z$ defined by $f\left(x\right)=2x$

Q. Which among the following functions are invertible?

1. $f:Z\to Z$ defined by $f\left(x\right)=x+2$
2. $f:Z\to Z$ defined by $f\left(x\right)=x^3$
3. $f:Z\to Z$ defined by $f\left(x\right)=|x|$
4. $f:Z\to Z$ defined by $f\left(x\right)=2x$

Q. Show that the Signum Function f : R → R , given by is neither one-one nor onto.

Q.

Choose the correct answer.

Let, where 0 ≤ θ≤ 2π, then

A. Det (A) = 0

B. Det (A) ∈ (2, ∞)

C. Det (A) ∈ (2, 4)

D. Det (A)∈ [2, 4]