Let a function ƒ : N →N be defined by
\(f(n) = \left\{ \begin{array}{ll} 2n & n = 2,4,6,8,\ldots \\ n - 1 & n = 3,7,11,15,\ldots \\ \frac{n+1}{2} & n = 1,5,9,13 \end{array} \right.\)
then, ƒ is
Let a function ƒ : N →N be defined by
\(f(n) = \left\{ \begin{array}{ll} 2n & n = 2,4,6,8,\ldots \\ n - 1 & n = 3,7,11,15,\ldots \\ \frac{n+1}{2} & n = 1,5,9,13 \end{array} \right.\)
then, ƒ is
- One-one but not onto
- Onto but not one-one
- Neither one-one nor onto
- One-one and onto
The Correct Option is D
Solution and Explanation
The correct answer is (D) : One-one and onto
For n=1,5,9,13 n=1,5,9,13, \(\frac{n+1}{2}\) yields all odd numbers.
When n=3,7,11,15,…n=3,7,11,15,…, n-1 is even but not divisible by 4.
For n=2,4,6,8,… n=2,4,6,8,…, 2n gives all multiples of 4.So, the range will be the set of all natural numbers.
Additionally, each value of n corresponds to a unique y, implying the function is one-to-one and onto.
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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