Question

Let $A=\\{1,2,3,4,5\\}, B=N$ and $f: A \rightarrow B$ be defined by f $(x)=x^{2}$
Find the range of $f$. Identify the type of function.

Answer 1

Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. A function is a relation for which each value from the set the first components of the ordered pairs is associated with exactly one value from the set of second components of the ordered pair.
A function is a relation in which each input has only one output. In the relation, $\mathrm{y}$ is a function of $\mathrm{x},$ because for each input $\mathrm{x}(1,2,3,$ or 0$),$ there is only one output $\mathrm{y} . \mathrm{x}$ is not a function of y, because the input $y=3$ has multiple outputs: $x=1$ and $x=2$

Complete step by step solution:
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain. A one-to-one function is a function of which the answers never repeat. For example, the function $f(x)=x+1$ is a one-to-one function because it produces a different answer for every input. An easy way to test whether a function is one-to one or not is to apply the horizontal line test to its graph.
Many-one Function: If any two or more elements of set A are connected with a single element of set $\mathrm{B}$, then we call this function as Many one function. Onto function or Surjective function : Function $\mathrm{f}$ from set $\mathrm{A}$ to set $B$ is onto function if each element of set $B$ is connected with set of A elements.
A function is said to be one-to-one if every y value has exactly one x value mapped onto it, and many-to-one if there are y values that have more than one x value mapped onto them. This graph shows a many-to-one function. The three dots indicate three x values that are all mapped onto the same y value.
$f(1)=1^{2}=1$
$\mathrm{f}(2)=2^{2}=4$
$\mathrm{f}(3)=3^{2}=9$
$f(4)=4^{2}=16$
$\mathrm{f}(5)=5^{2}=25$

Hence, the range of the function is {1, 4, 9, 16, 25} since the elements in the co-domain do not equal to the elements in range $\Rightarrow \mathrm{f}$ is not onto.
Image is not the same for two domain elements in $\mathrm{f}$, hence it is one-one.

Note:
Types of Functions
- One - one function (Injective function)
- Many - one function.
- Onto - function (Surjective Function)
- Into - function.
A function is an equation that has only one answer for $y$ for every $x$. A function assigns exactly one output to each input of a specified type. It is common to name a function either $f(x)$ or $g(x)$ instead of $y . f(2)$ means that we should find the value of our function when $x$ equals 2 .
A function is a generalized input-output process that defines a mapping of a set of input values to a set of output values. A student must perform or imagine each action. A student can only imagine a single value at a time as input or output (e.g., x stands for a specific number).
A function is surjective or onto if each element of the codomain is mapped to by at least one element of the domain. In other words, each element of the codomain has non-empty preimages. Equivalently, a function is surjective if its image is equal to its codomain. A surjective function is a surjection.