What Are Functions in Math?

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.

Representation-

Functions are generally represented as \(f(x)\)

Let , \(f(x)= x^{3}\)

It is said as f of x is equal to x cube.

Functions can also be represented by g(), t(),… etc.

Lets Work Out- Example- Find the output of the function \(g(t)= 6t^{2}+5\) (i) t = 0 (ii) t = 2 Solution- The given function is \(g(t)= 6t^{2}+5\) (i) At t = 0, \(g(0)= 6(0)^{2}+5 \) = 5 (ii) At t = 2, \(g(2)= 6(2)^{2}+5 \) = 29 |

Types of Functions-

One-to-One function – It signifies that one element of a set is related to one element of the other set.

Many-to-One-function- It signifies that many elements of a set is related to one element of the other set.

Note : One-to-Many- A function cannot have a one to many relation ie. one element of a set cannot be related to more than one element of the other set. |

Vertical Line Test-

Vertical line test is used to determine whether a curve is a function or not.

If any curve cuts a vertical line at more than one points then the curve is not a function.

Types of Functions-

Polynomial function-

A polynomial function can be expressed as :

\(f(x)= a_{n}x^{n}+a_{n-1}x^{n-1}+…..+a_{1}x^{1}+a_{0}\)

The highest power in the expression is known as the degree of the polynomial function. The different types of polynomial functions based on the degree are:

- The polynomial function is called as Constant function if the degree is zero.
- The polynomial function is called as Linear if the degree is one.
- The polynomial function is Quadratic if the degree is two.
- The polynomial function is Cubic if the degree is three.

Constant Polynomial Functions

The polynomial of 0th degree where f(x) = f(0) = \(a_{0}\)

Linear Polynomial Functions

A linear polynomial function is a first degree polynomial where the input needs to be multiplied by m and added to c. It can be expressed by f(x) = mx + c.

For example, f(x) = 2x + 1 at x = 1

f(1) = 2.1 + 1 = 3

f(1) = 3

Quadratic Polynomial Functions

A Quadratic polynomial function is a second degree polynomial and it can be expressed as;

F(x) = ax 2 + bx + c, and a is not equal to zero.

Where a, b, c are constant and x is a variable.

Example, f(x) = 2x 2 + x – 1 at x = 2

If x = 2, f(2) = 2.2 2 + 2 – 1 = 9

Cubic Polynomial Function

A cubic polynomial function is a polynomial of degree three and can be expressed as;

F(x) = ax 3 + bx 2 + cx + d and a is not equal to zero.

Algebraic Functions

A function that consists of a finite number of terms involving powers and roots of independent variable x and fundamental operations such as addition, subtraction, multiplication, and division is known as an algebraic equation.

For Example-

\(f(x)=5x^{3}-2x^{2}+3x+6\)

Rational functions, irrational functions and Polynomials functions are examples of algebraic functions.

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