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 \(g(t)= 6t^{2}+5\) at (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) = \(a_{0}\)=c. Regardless of the input, the output always result in constant value. The graph for this is a horizontal line.
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) = ax2 + bx + c, and a is not equal to zero.
Where a, b, c are constant and x is a variable.
Example, f(x) = 2x2 + x – 1 at x = 2
If x = 2, f(2) = 2.22 + 2 – 1 = 9
Cubic Polynomial Function
A cubic polynomial function is a polynomial of degree three and can be expressed as;
F(x) = ax3 + bx2 + 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-
\(f(x)=5x^{3}-2x^{2}+3x+6\), \(g(x)=\frac{\sqrt{3x+4}}{(x-1)^{2}}\).
Rational functions, irrational functions and Polynomials functions are examples of algebraic functions.
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