# Functions

 Question 1
Which one of the following is the closed form for the generating function of the sequence $\{a_n \}_{n \geq 0}$ defined below?
$a_n= \left \{ \begin{matrix} n+1, &n \text{ is odd} \\ 1,& \text{otherwise} \end{matrix} \right.$
 A $\frac{x(1+x^2)}{(1-x^2)^2}+\frac{1}{1-x}$ B $\frac{x(3-x^2)}{(1-x^2)^2}+\frac{1}{1-x}$ C $\frac{2x}{(1-x^2)^2}+\frac{1}{1-x}$ D $\frac{x}{(1-x^2)^2}+\frac{1}{1-x}$
GATE CSE 2022   Discrete Mathematics
Question 1 Explanation:
 Question 2
Consider the following sets, where $n\geq 2$:

S1: Set of all nxn matrices with entries from the set $\{a, b, c\}$
S2: Set of all functions from the set$\{0,1,2,...,n^2-1\}$ to the set $\{0, 1, 2 \}$

Which of the following choice(s) is/are correct?
[MSQ]
 A There does not exist a bijection from S1 to S2 B There exists a surjection from S1 to S2 C There exists a bijection from S1 to S2 D There does not exist an injection from S1 to S2
GATE CSE 2021 SET-2   Discrete Mathematics
Question 2 Explanation:
 Question 3
If the ordinary generating function of a sequence $\{a_{n}\}_{n=0}^{\infty} \; is \; \frac{1+z}{(1-z)^{3}}$ then $a_{3}-a_{0}$ is equal to ______.
 A 16 B 17 C 15 D 14
GATE CSE 2017 SET-2   Discrete Mathematics
Question 3 Explanation:
 Question 4
If $g(x)=1-x$ and $h(x)=\frac{x}{x-1}$, then $\frac{g(h(x))}{h(g(x))}$ is:
 A $\frac{h(x)}{g(x)}$ B $\frac{-1}{x}$ C $\frac{g(x)}{h(x)}$ D $\frac{x}{(1-x)^{2}}$
GATE CSE 2015 SET-1   Discrete Mathematics
Question 4 Explanation:
 Question 5
Let A be a finite set having $x$ elements and let B be a finite set having $y$ elements. What is the number of distinct functions mapping B into A.
 A $x^{y}$ B $2^{(x+y)}$ C $y^{x}$ D $y ! /(y-x) !$
ISRO CSE 2014   Discrete Mathematics
Question 5 Explanation:
 Question 6
Consider the set of all functions f:{0,1,...,2014}$\rightarrow${0,1...,2014} such that f(f(i))=i, for 0$\leq$i$\leq$2014 . Consider the following statements.
P. For each such function it must be the case that for every i, f(i) = i,
Q. For each such function it must be the case that for some i,f(i) = i,
R. Each such function must be onto.
Which one of the following is CORRECT?
 A P, Q and R are true B Only Q and R are true C Only P and Q are true D Only R is true
GATE CSE 2014 SET-3   Discrete Mathematics
Question 6 Explanation:
 Question 7
Let S denote the set of all functions f:${\{0,1\}}^{4} \rightarrow \{0,1\}$. Denote by N the number of functions from S to the set {0,1}. The value of $log_{2} log_{2}N$ is______.
 A 4 B 8 C 16 D 32
GATE CSE 2014 SET-1   Discrete Mathematics
Question 7 Explanation:
 Question 8
How many onto (or surjective) functions are there from an n-element ( $n \geq 2$ ) set to a 2-element set?
 A $2^{n}$ B $2^{n}-1$ C $2^{n}-2$ D $2(2^{n}-2)$
GATE CSE 2012   Discrete Mathematics
Question 8 Explanation:
 Question 9
Let $f$ be a function from a set A to a set B, $g$ a function from B to C, and $h$ a function from A to C, such that $h(a) = g(f(a))$ for all $a \in A$. Which of the following statements is always true for all such functions $f$ and $g$?
 A g is onto $\implies h$ is onto B h is onto $\implies f$ is onto C h is onto $\implies g$ is onto D h is onto $\implies f$ and g are onto
GATE IT 2005   Discrete Mathematics
Question 9 Explanation:
 Question 10
Let f: B $\rightarrow$ C and g: A $\rightarrow$ B be two functions and let h = f$\cdot$g. Given that h is an onto function which one of the following is TRUE?
 A f and g should both be onto functions B f should be onto but g need to be onto C g should be onto but f need not be onto D both f and g need to be onto
GATE CSE 2005   Discrete Mathematics
Question 10 Explanation:
There are 10 questions to complete.

### 2 thoughts on “Functions”

1. Q1 is from calculus, kindly move it from here to the calculus section