# Why are Functions Important in Mathematics?

In mathematics, the functions are used to identify the varying quantity with respect to other quantity. It is represented as f(x). In other words, the function is a relation in which each element in a domain is exactly associated with one element of a co-domain. Functions have various inputs and outputs which are represented in many ways. They are:

- Numeric
- Verbal
- Algebraic
- Graphical

The functions in mathematics are important because the theories about dependencies between two quantities are developed. There are different **types of functions** in mathematics such as

- Algebraic function
- Constant function
- Polynomial function
- Linear function
- Logarithmic function
- Quadratic function
- Cubic Function
- Trigonometric function

#### Algebraic Function

The function which is defined using algebraic expressions is called algebraic functions.

Example: f(x) = x^{2} + x + 2

#### Constant Function

If X and Y are the two non-empty sets, then the function from X to Y is called constant function *iff *the range of “f” is a singleton set.

#### Polynomial Function

The polynomial function of degree “n” is of the form P(X) = a_{n}x^{n} + a_{n-1}x^{n-1}+….+a_{1}x +a_{0} where “n” is a positive integer and a_{n}, a_{n-1},….a_{1}, a_{0 }are the real numbers.

Example: x^{2} – 4x + 7

#### Linear Function

The linear function is a polynomial function that has one independent variable and one dependent variable. The general form of a linear function is

f(x) = ax +b

Example: y = 25 + 5x

#### Logarithmic Function

The function which appears as an argument of a **logarithm** is called logarithmic function. The general form of a logarithmic function is

f(x) = log_{a}(x)

Example: f(x) = log sin (x)

#### Quadratic Function

The quadratic function is defined as the polynomial equation of degree “2” and is of the form

f(x) = ax^{2} + bx + c

Example: x^{2 }+ 5x + 6

#### Cubic Function

The polynomial function with highest degree of “3” is called cubic function with general form

f(x) = ax^{3} +bx^{2}+cx+d

Example: 2x^{3} +3x – 6

#### Trigonometric Function

The function in which an angle is represented as the ratio of two sides of a right-angled triangle is called trigonometric functions.

Example: f(x) = sin x, f(x)= cos x, f(x) = tanx etc.

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