Let $N$ be the set of natural numbers and two functions $f$ and $g$ be defined as $f,g : N \to N$ such that :
$f(n) = \begin{cases}
\frac{n+1}{2} & \quad \text{if } n \text{ is odd}\\
\frac{n}{2} & \quad \text{if } n \text{ is even}
\end{cases}$
and $g(n) = n-(-1)^n$. The $fog$ is :
- Both one-one and onto
- One-one but not onto
- Neither one-one nor onto
- onto but not one-one
The Correct Option is D
Solution and Explanation
$g(x) = n - (-1)^n \begin{cases} n+ 1 & ; \quad n \text{ is odd}\\ n - 1 &; \quad \text{if } n \text{ is even} \end{cases} $
$f(g(n)) = \begin{cases} \frac{n}{2} ; & \quad n \text{ is even}\\ \frac{n +1}{2} ; & \quad n \text{ is odd} \end{cases} $
$\therefore$ many one but one to one
Top Questions on Functions
- If $ f(x) = 2x^2 - 3x + 5 $, find $ f(3) $.
- The domain of the real valued function $ f(x) = \frac{3}{4 - x^2} + \log_{10}(x^3 - x) $ is
- If $ 0 \le x \le 3,\ 0 \le y \le 3 $, then the number of solutions $(x, y)$ for the equation: $$ \left( \sqrt{\sin^2 x - \sin x + \frac{1}{2}} \right)^{\sec^2 y} = 1 $$
- A real valued function $ f: A \to B $ defined by $ f(x) = \frac{4 - x^2}{4 + x^2} \ \forall x \in A $ is a bijection. If $ -4 \in A $, then $ A \cap B = $
- Let the function $ f(x) = \sqrt{\log_e(1 - x^2)} $. Then the domain of $ f(x) $ is:
Questions Asked in JEE Main exam
- A force of 49 N acts tangentially at the highest point of a sphere (solid of mass 20 kg) kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is:
- JEE Main - 2025
- Quantum Mechanics
Let \( A = \{-3, -2, -1, 0, 1, 2, 3\} \). A relation \( R \) is defined such that \( xRy \) if \( y = \max(x, 1) \). The number of elements required to make it reflexive is \( l \), the number of elements required to make it symmetric is \( m \), and the number of elements in the relation \( R \) is \( n \). Then the value of \( l + m + n \) is equal to:
- JEE Main - 2025
- Sets and Relations
For hydrogen-like species, which of the following graphs provides the most appropriate representation of \( E \) vs \( Z \) plot for a constant \( n \)?
[E : Energy of the stationary state, Z : atomic number, n = principal quantum number]- JEE Main - 2025
- Hydrogen
- Concentrated nitric acid is labelled as 75%by mass. The volume in mL of the solution which contains 30 g of nitric acid is: Given: Density of nitric acid solution is 1.25 g/mL.
- JEE Main - 2025
- Solutions
The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is $ 4 \_\_\_\_\_$.
- JEE Main - 2025
- permutations and combinations
JEE Main Notification
Concepts Used:
Functions
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets, mapping from A to B will be a function only when every element in set A has one end only one image in set B.
Kinds of Functions
The different types of functions are -
One to One Function: When elements of set A have a separate component of set B, we can determine that it is a one-to-one function. Besides, you can also call it injective.
Many to One Function: As the name suggests, here more than two elements in set A are mapped with one element in set B.
Moreover, if it happens that all the elements in set B have pre-images in set A, it is called an onto function or surjective function.
Also, if a function is both one-to-one and onto function, it is known as a bijective. This means, that all the elements of A are mapped with separate elements in B, and A holds a pre-image of elements of B.
Read More: Relations and Functions