(C) Testing for one-one: $f(1) = \frac{1+1}{2} = 1$ and $f(2) = \frac{2}{2} = 1$. Since $f(1) = f(2)$ for $1 \neq 2$,the function is not one-one (it i

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(C) Testing for one-one: $f(1) = \frac{1+1}{2} = 1$ and $f(2) = \frac{2}{2} = 1$. Since $f(1) = f(2)$ for $1 \neq 2$,the function is not one-one (it is many-one).Testing for onto: For any $y \in N$,if $y$ is odd,we can have $n = 2y-1$,then $f(2y-1) = \frac{(2y-1)+1}{2} = y$. If $y$ is even,we can have $n = 2y$,then $f(2y) = \frac{2y}{2} = y$. Thus,for every element in the codomain,there is a pre-image in the domain. Therefore,the function is onto.

Class 12 Mathematics Relation and Function Type of Functions based on Mapping

Medium

Let $f: N \rightarrow N$ be defined by $f(n) = \begin{cases} \frac{n+1}{2}; & \text{if } n \text{ is odd} \\ \frac{n}{2}; & \text{if } n \text{ is even} \end{cases}$,for all $n \in N$ then $f$ is $\dots \dots \dots$

GUJCET 2026 All GUJCET

A
One-one but not onto
B
One-one and onto
C
Many-one and onto
D
Neither one-one nor onto

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Relation and Function

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Type of Functions based on Mapping

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In each of the following cases,state whether the function is one-one,onto or bijective. Justify your answer. $f : R \rightarrow R$ defined by $f(x) = 1 + x^2$.

Medium

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The function $f(x) = \sin (\log (x + \sqrt {x^2 + 1}))$ is

Medium

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Let $A = \{x_1, x_2, \dots, x_7\}$ and $B = \{y_1, y_2, y_3\}$ be two sets containing seven and three distinct elements respectively. The total number of onto functions $f : A \to B$ such that there exist exactly three elements $x$ in $A$ with $f(x) = y_2$ is equal to:

DifficultJEE MAIN 2015

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Show that the function $f : R \rightarrow \{ x \in R : -1 < x < 1 \}$ defined by $f(x) = \frac{x}{1+|x|}$ for all $x \in R$ is a one-one and onto function.

Easy

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Let $f: N \rightarrow N$ be defined by $f(n) = \begin{cases} \frac{n+1}{2}; & \text{if } n \text{ is odd} \\ \frac{n}{2}; & \text{if } n \text{ is even} \end{cases}$. Then $f$ is:

DifficultKCET 2014

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Let $f: N \rightarrow N$ be defined by $f(n) = \begin{cases} \frac{n+1}{2}; & \text{if } n \text{ is odd} \\ \frac{n}{2}; & \text{if } n \text{ is even} \end{cases}$,for all $n \in N$ then $f$ is $\dots \dots \dots$

Similar Questions

In each of the following cases,state whether the function is one-one,onto or bijective. Justify your answer. $f : R \rightarrow R$ defined by $f(x) = 1 + x^2$.

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Let $A = \{x_1, x_2, \dots, x_7\}$ and $B = \{y_1, y_2, y_3\}$ be two sets containing seven and three distinct elements respectively. The total number of onto functions $f : A \to B$ such that there exist exactly three elements $x$ in $A$ with $f(x) = y_2$ is equal to:

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