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
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.
A functionis 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.
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.
