Types of Functions
The true meaning of functions is that they are the relations at the place of which each input has a specific output.
‘Functions and its types’ chapter explains the basic concepts, as well as, the various types of function in math. All are described below using several examples for better understanding.
The very definition of a function is: it is an association between a set of inputs and a set of allowable outputs with the ability that each input is linked to exactly one output.
Here you can learn about various types of functions, as well as the types of functions graphs, in mathematics, in brief.
Types of Functions in Mathematics
The types of functions in Mathematics are:
One to One Function
Many to One Function
Onto Function
One - One and Onto Function
Identity Function
Constant Function
Polynomial Function
Quadratic Function
Rational Function
Modulus Function
Signum Function
Greatest Integer Function
Different Types of Functions
The different types of functions explain various mathematical relations and their graphical representation.
Identity Function
Consider R as the set of real numbers. The function is said to be the Identity function, if f: R→R is explained as f(x) = y = x, for x ∈ R, the domain and the range being R. The types of functions in sets can be understood with the help of these functions.
The Identity Function graph is a straight line and always passes through the origin.
[Image will be uploaded soon]
Constant Function
Here, the condition about the function f: R→R is distinct as f(x) = y = c, for x ∈ R and c is a constant in R, at this point such a function is identified as Constant function. R is The domain of the function (f), and its range is a constant (c).
By plotting the graph, we discover a straight line parallel to the x-axis.
[Image will be uploaded soon]
Polynomial Function
A polynomial function is explained by y = a0 + a1x + a2x2 + … + anxn, here ‘n’ is a non-negative integer and a0, a1, a2,…, n ∈ R. The uppermost power in the equation is the degree of the polynomial function.
Also, these polynomial functions are categorized as per their degrees:
Constant function: The polynomial function will be constant if the degree is zero.
Linear function: This kind polynomial function possesses the degree ‘one’. For instance;
y = x + 1 or y = x or y = 2x – 5, etc.
If we consider the equation, y = x – 6. R is the domain as well as the range. The graph is a straight line.
[Image will be uploaded soon]
Quadratic Function
A function is said to be a quadratic function if the degree of the polynomial function is two. This can be represented as f(x) = ax2 + bx + c, where a ≠ 0 and a, b, c is constant, and x is a variable. R is the domain and the range. The plotting in graph of a quadratic function for the function: f(x) = x2 – 4 is
[Image will be uploaded soon]
Rational Function
A rational function can be characterized such as, f(x) / g(x) where both the f(x) and g(x) are polynomial functions of x, at a condition g(x) ≠ 0.
If f: R → R and it says that f(x) = 1 / (x + 2.5).
The Graphical representation shows asymptotes, the curves which seem to touch the axes-lines.
[Image will be uploaded soon]
Modulus Function
The outright value of any number, c is characterized in the form of |c|. It can be said that f: R→ R is well-defined by f(x) = |x|.
f(x) = x: for each non-negative value of x, and
f(x) = - x: for each negative value of x
For example
f(x) = {- x, if x < 0
= {– x, if x ≥ 0.
[Image will be uploaded soon]
Signum Function
f: R→ R can be explained as
f(x) = {0, if x = 0
= {1, when x > 0
= {-1, if x < 0
[Image will be uploaded soon]
Greatest Integer Function
The condition for a function f: R→ R is denoted by f(x) = [x]; such that x ∈ X. It completes the real number to the integer less than the number. The types of functions in sets become easier to comprehend with the help of respective graphs.
Assume that the interval is in the shape of (k, k+1), the utmost integer value of the function is k, which is an integer.
For illustration:
[-21] = 21, [5.12] = 5. The graphical demonstration is
[Image will be uploaded soon]
Types of Function Graphs
Here you can learn about the different types of functions and their graphs easily by referring to the illustrations.
Some common types of function graphs are shown below.
[Image will be uploaded soon]
Types of Functions in Sets
There are various types of functions that are propagated in sets. These types of functions in sets are discussed below.
i. One-to-One Function
The condition for a function f: A → B to be One to One when each element of A is an individual element of B.
[Image will be uploaded soon]
ii. Many-to-One Function
This kind of function plots two or more elements of A to the identity element of set B.
[Image will be uploaded soon]
iii. Onto Function
A function is called onto function, which represents that each element of set B is (are) preimage(s) in set A.
[Image will be uploaded soon]
iv. One - One and Onto Function
This function is possible when each element of A is a distinctive element of B.
[Image will be uploaded soon]
FAQs on Functions and its Types
Q1: Explain the working procedure of a Function.
Ans: A function is a binary relation between two sets that associates every element of the first set to exactly one element of the second set. Otherwise, a function denotes an equation that delivers only one answer for ‘y’ for every value of ‘x’.
This is the best chance to get the precise one output to a respective input of an identified type by a function.
Q2: How can one determine if an equation is a Function?
Ans: One can determine whether an equation is a function by solving for y.
When you solve an equation with a specific value for x, there shall be only one corresponding y-value for that particular x-value.
Q3: Can you identify a Function as given in the following figures?
[Image will be uploaded soon]
Ans: (iii)
Yes, the number (iii) is an illustration of a function. As the specified function draws every element of A with that of B.
In number (ii), the specified function draws one element of A with two elements of B (one to many).
In number (i), there is a defilement of the definition of the function as the assumed function does not draw every element of A.
Q4: What is your opinion about a parent Function?
Ans: It is the modest function that always gratifies the meaning of a definite type of function. For illustration, at the time we consider the linear functions which structure a family of functions, then y = x would be the parent function. This is the modest linear function.
