There is an important branch of mathematics that deals with different types of sets and operations on them, known as **set theory**. Relations and functions are very commonly studied in set theory. There are certain sets (usually two) that are connected with one another by means of some property. This is termed as a **relation**. The **function** is an extension of the relation. A function is defined as a type of relation in which every element of one set is connected to an element of other. A relation is known as a function if for each input value, there exists a single output value.

Function is notified usually by **f, F **and sometimes by **g** and **$\phi$** too. If function f shows a relation between two sets A and B, then it may be represented by -**f : A $\rightarrow$ B**

Where, A is mapped on B.

Function notation means to find the exact value of the function. An equation will be a function if for any independent variable in the domain of the equation the equation will yield exactly one value of the dependent variable. Solving functions in math deal with finding unknown variable from the given expression with the help of known values.

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When we use a notation f(x), we are using the function notation. The purpose of using the function notation is that we can write more information instead of using symbols, we can use just the variable y. For example, we ask how much money a person will make for working 21 hours is simply a matter of asking for f(21). To indicate that y is the function of x. We often use a function notation and write y = f(x). We read it as "y is a function of x."

The notation F:A→B indicates that 'F' is a function with the domain A and the co-domain B , and the function 'F' is said to mapping or associate elements of A to elements of B. In this case, the set of all elements of B is called as the image of the function. The term range usually refers to the image but sometimes it known as the co-domain.

A specific input in a function notation is called an argument of the function notation. For each argument value of a, the corresponding unique value of b in the co-domain is called as the function notation value at 'a', output of F for an argument 'a' or the image of 'a' under F. The image of 'a' may be can be written as F(a) or as 'b'.### Solved Example

**Question: **Find the value of f(3) in the equation f(x) = 2x^{2}+18x + 4.

** Solution: **
The function notation of F(X) is explained as follows:

Consider the function of notation f(x) = 2x + 1

Substitute the value x = a

Here, f(a) is the function, where 'a' is the input.

So, f(a) = 2a + 1 is the final solution.

### Solved Examples

**Question 1: **Find f(2) for the function f(x) = 2x + 3

** Solution: **

**Question 2: **f (x) = x^{2} + 2x + 12. Find the value when x = 5

** Solution: **

**Question 3: **Find n(3) for the function n(x) = x^{3} + 3x + 2

** Solution: **
**Question 4: **Find m(3) for the function m(x) = x^{3} + 2x^{2} + 2x + 36

** Solution: **
**Question 5: **Find n(7) for the function n(x) = x^{3} + 2x^{2} + 2x + 54

** Solution: **
We know that any expression in x and whose values depends on x is called
the function. If all the values of x are real numbers, then we can say
that the given function is step function. In general, we have two types
of step functions as "Greatest Integer function" and "Smallest Integer
function". These types of function can be expressed in the steps and we
can use $\left [ [] \right ]$ symbol to represents the step function.

For example, $[ [x + 3] ]$, $[ [x^{2} + 1] ]$ are called as the step functions. In calculus, we have unit step function, heaviside step function etc.

Given below are some of the examples that shows us how to graph a function.### Solved Example

**Question: **Graph the function y = f(x) = x^{2} - 1.

** Solution: **
If we have a function f = { (1,2), (2,3) ,(-1,4)} and if we interchange
the first and second elements of each of the ordered pairs then we get
the set {(2,1), (3,2), (4,-1)} and is called the inverse function of f. It is denoted by f^{-1} and represented as f^{-1} = {(2,1), (3,2), (4,-1)}
In composite function, we combined two functions under a given formula, in
which we can apply first function and get the answer and after
that use this answer in to the second function.

If $f : X \rightarrow Y$ and $g: Y \rightarrow Z$, then composite function of these two can be expressed as $g o f$ or $g[f(x)]$. Then, we have $g o f: X \rightarrow Z$ and co-domain of $f$ is the domain of $g$.

The notation F:A→B indicates that 'F' is a function with the domain A and the co-domain B , and the function 'F' is said to mapping or associate elements of A to elements of B. In this case, the set of all elements of B is called as the image of the function. The term range usually refers to the image but sometimes it known as the co-domain.

A specific input in a function notation is called an argument of the function notation. For each argument value of a, the corresponding unique value of b in the co-domain is called as the function notation value at 'a', output of F for an argument 'a' or the image of 'a' under F. The image of 'a' may be can be written as F(a) or as 'b'.

The value of x is given as 3

f(3) = 2(3^{2}) + 18(3) + 4

f(6) = 18 + 54 + 4

f(6) = 76

f(3) = 2(3

f(6) = 18 + 54 + 4

f(6) = 76

- F - Function of variable and it has to identify the term of the specification.
- X - It is the notation for the given function. It may be any numbers or character from their input function.

Consider the function of notation f(x) = 2x + 1

Substitute the value x = a

Here, f(a) is the function, where 'a' is the input.

So, f(a) = 2a + 1 is the final solution.

Given below are some of the examples on function notation.

f(x) = 2x + 3

Substitute the value given in the question

So, substitute the value x = 2 in equation

f(2) = 2(2) + 3

f(2) = 4 + 3

f(2) = 7

f(x) = x^{2} + 2x + 12

Substitute the value given in the equation.

So, substitute the value x = 5 in the equation.

f(5) = 5^{2} + 2(5) + 12

= 25 + 10 +12/p>

= 47

n(x) = x^{3} + 3x + 2

n(3) = (3^{2}) + 3(3) + 2

n(3) = 9 + 9 + 2

n(3) = 20

n(3) = (3

n(3) = 9 + 9 + 2

n(3) = 20

m(x) = x^{3} + 2x^{2 }+ 2x + 36

m(3) = (3^{3}) + 2(3^{2}) + 2(3) + 36

m(3) = 27 + 18 + 6 + 36

m(3) = 87

m(3) = (3

m(3) = 27 + 18 + 6 + 36

m(3) = 87

n(x) = x^{3} +2x^{2 }+ 2x + 54

n(7) = (7^{3})+ 2(7^{2}) + 2(7) +54

n(7) = 343 + 98 + 14 + 54

n(7) = 509

n(7) = (7

n(7) = 343 + 98 + 14 + 54

n(7) = 509

For example, $[ [x + 3] ]$, $[ [x^{2} + 1] ]$ are called as the step functions. In calculus, we have unit step function, heaviside step function etc.

Given below are some of the examples that shows us how to graph a function.

First, we have to find the values of y with the corresponding values of x as follows:

x = 0, then y = -1

x = 1, then y = 0

x = -1, then y = 0

x = 2, then y = 3

x = -2, then y = 3

x = 1, then y = 7

x = -3, then y = 3

By the use of these points, we can graph the function notation as follows:

It is clear from the graph that y = 0 for x = 1 and x = -1.

x = 0, then y = -1

x = 1, then y = 0

x = -1, then y = 0

x = 2, then y = 3

x = -2, then y = 3

x = 1, then y = 7

x = -3, then y = 3

By the use of these points, we can graph the function notation as follows:

It is clear from the graph that y = 0 for x = 1 and x = -1.

If $f : X \rightarrow Y$ and $g: Y \rightarrow Z$, then composite function of these two can be expressed as $g o f$ or $g[f(x)]$. Then, we have $g o f: X \rightarrow Z$ and co-domain of $f$ is the domain of $g$.

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