Functions

Introduction

A function in mathematics is a relationship between a group of inputs with one output. A function, in simple terms, is a relationship between inputs in which each input is associated with only one output. Each function has a domain and a co-domain, often known as a range. The general notation is f(x), where x represents the input. A function’s general representation is y = f(x).

Here, we’ll learn about these functions in mathematics and their various types.

What is a Function in Mathematics?

In mathematics, a function is a specific relationship between inputs (the domain) and outputs (the co-domain). Each input has exactly one output, and the output can be traced back to its input.

Types of Functions in Math

F(x) = x^2 is an example of a basic function. The function f(x) takes the value of “x” and squares it in this function. If x = 3, for instance, f(3) = 9. 

f(x) = sin x, f(x) = x^2+ 3, f(x) = 1/x, f(x) = 2x + 3, and so on are some additional instances of functions.

There are several types of functions in math. Some important types are:

• Injective function or one to one function: The domain of a function is considered an injective function if each element in the domain has a distinct image in the co-domain. There is a mapping between two sets for a range in each domain

• Surjective functions or onto function: When all the elements in a co-domain are mapped from a domain, i.e. co-domain is equal to the range, it is a surjective function

If f: A→B is one to one, there is at least one element ‘a’ in the domain such that f(a) = b, i.e., the function maps one or more elements of A to the same element of B for every element ‘b’ in the co-domain B.

• Polynomial function: A polynomial function is a four-operation mathematical function made up of constants and variables. Only non-negative integer powers of variables are used in equations, and the polynomial function is quadratic, cubic, or bi-quadratic

Graphical Representation of Polynomial Function:

When we graphically display a polynomial function, the horizontal axis touches zero and with even multiples. It also changes direction at the turning point, i.e., a polynomial function with degree ‘n’ has at most n-1 turning points. The graph of a polynomial function is smooth, curved, and continuous.

• Smooth curved: Smooth curved polynomial functions of degree two or more have a graph with no sharp corners

• Continuous: The curve with no break is called “continuous” when the polynomial graph has no break

• Inverse Functions: Alternatively known as an anti-function, an inverse function can be inverted into an extra function. In other words, if a function “f” converts x to y, the inverse of the same function will convert y to x

These were a few function samples. It’s worth noting that there are a variety of additional functions available, such as the intro function, algebraic functions, and so on.

How to Find the Domain of a function?

• To determine the domain, we must examine the values of the independent variables, which are permitted to be used as mentioned before, i.e., no negative sign within the square root and no “0” sign at the bottom of the fraction

• In general, the domain of a function is defined as the set of all real numbers (R), subject to various limitations. They are as follows:

• The domain is “the set of all real numbers” when the specified function is of the kind f(x) = 2x + 5 or f(x) = x^2– 2

• The domain is the set of all real numbers except 1 when the supplied function is of the form f(x) = 1/(x – 1)

• Domain limitations refer to the range of values for which a function in math cannot be defined

How to Find the Range of a Function

The range of the function is the spread of all the y values from least to maximum.

Substitute all the values of x in the supplied expression of y to see if it is positive, negative, or equal to other values.

Determine y’s minimum and maximum values.

Then create a graph for it.

“It is feasible to restrict the range (i.e., the output of a function) by redefining the co-domain of that function,” according to one intriguing fact concerning range and co-domain. The set of all positive integers, negative real numbers, and so on must be the co-domain of f(x). The function’s output must be a positive integer in this scenario, and the domain will be restricted as well.

We’ve used upper-case letters to represent functions so far, but lower-case characters are more common. If (a,b) f and f(a) = b, then f(a) = b. The image of an under f is referred to as b, while the preimage of b under f is referred to as a.

Conclusion 

We’ve learned about the function in math, injective function, surjective function, polynomial function, the function’s range, the function’s domain, and more. There’s no disputing that the subject is extensive and requires a significant amount of time to comprehend.