The function y = f(x) is classified into different types of functions, based on factors such as the domain and range of a function, and the function expression. The functions have a domain x value that is referred as input. The domain value can be a number, angle, decimal, fraction. Similarly, the y value or the f(x) value (is generally a numeric value) is the range. The types of functions have been classified into the following four types.
There are three different forms of representation of functions. The functions need to be represented to showcase the domain values and the range values and the relationship between them.
The types of functions are classified further to help for easy understanding and learning. The types of functions have been further classified into four different types, and are presented as follows.
1. Types of Functions  Based on Set Elements
These types of functions are classified based on the number of relationships between the elements in the domain and the codomain. The different types of functions based on set elements are as follows.
2. Types of Function  Based on Equation
Identity Function
Linear Function
Quadratic Function
Cubic Function
Polynomial Function
3. Types of Functions  Based on Range
Here the types of functions have been classified based on the range which is obtained from the given functions. The different types of functions based on the range are as follows.
4. Types oF Functions  Based on Domain
Functions are used in all the other topics of maths. The functions have been classified based on the types of equations used to define the functions. The function equations generally have algebraic expressions, trigonometric functions, logarithms, exponents, and hence are named based on these domain values. The three broad types of functions based on domain value are as follows.
Example 1: For the given functions f(x) = 3x + 2 and g(x) = 2x  1, find the value of fog(x).
The given two functions are f(x) = 3x + 2 and g(x) = 2x  1.
We need to find the function fog(x).
fog(x) = f(g(x))
= f(2x1)
= 3(2x  1) + 2
= 6x  3 + 2
= 6x  1
Therefore fog(x) = 6x  1
Example 2: Find the inverse function of the function f(x) = 5x + 4.
The given function is f(x) = 5x + 4
we rewrite it as y = 5x + 4 and simplify it to find the value of x.
y = 5x + 4
y  4 = 5x
x = (y  4)/5
f^{1}(x) = (x  4)/5
Therefore the inverse function is f1(x) = (x  4)/5
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