Real-Valued Functions

Introduction

A function in mathematics is an assignment of an element to another element from a set to another set. A function from a set A to a set B is an assignment of an element of set B to each element of a set A. Set A is called the domain of the function, set B is called codomain, and the set of all the values relating from set A to set B is called the range. A real-valued function has its range as a set of all real numbers. No complex numbers are involved.

What is a Real-Valued Function?

Let us now know about real function and real-valued function. All those functions that have the range of all the actual values are called real-valued. They have certain aspects of mathematics. The range or image of the function is the set of values attained by the function.

For example, let A and B be two sets that are subsets of R (set of all real numbers). Function f: A⟶B is called a real-valued function if the range of f is also a set of all the real values. 

What is a Polynomial Function?

Let y = f(x) = 6x²+2x -34

In this polynomial expression, x is an independent variable as it can take any value, whereas y is a dependent variable because its value depends upon the value of x. Hence, the function in which the independent variable is a part of the polynomial expression is called a polynomial function. Some important points are:

• The domain of polynomial functions is always a set of real values.
• Polynomial functions are a real-valued function example.
• Same degree polynomial functions have similar characteristics.
• Polynomial functions of the type f(x) = y = anxn + an-1xn-1 + an-2xn-2 + … + a1x + a0 

where all the a are constants or real values and all n are non-negative integers.

What is a Linear Function?

The polynomial functions that have the degree zero or one are called linear functions. The graph of a linear function is a straight line. The example of this is clear and simple. For an instance, y = f(x) = 5x+2 is a linear function as the degree of the function is 1. The same goes for other functions of this type as well. It is the simplest form of a function, which is real-valued. Thus, we can conclude the following:

• A linear function has one independent and one dependent variable.
• The graph for a linear function is just a straight line. 
• Linear Functions are also real-valued and have all the properties that it follows.
• All linear functions follow a general structure in the form of y = a + bx where a and b are the coefficients and x has a maximum degree one.

What is an Identical Function? 

An identical function is an equal function. When we resolve and convert two functions into their standard forms, if they are equal, then they are called identical functions. For two functions to be identical, they should be equal. Although there is no particular structure that the identical functions follow, they are the simplest among them all. 

For example, if f(x) = 2 log x and b(x) = log x2, then b(x) can be further solved using the properties of logarithm and in its simplest form, the two functions f(x) and b(x) become equal and identical. Hence, we call them identical functions. Thus, the concept of identical functions is clear and simple. The main points of identical functions are as follows:

• Identical functions can include two or more functions that can be made equal and the same for getting identical.
• All the identical functions are also real-valued because they can also take all the real values.
• The domain and range of the identical functions are also the same and equal.
• When all the functions are simplified, and they become equal in expression or value, they become identical functions. They are also called equal functions.

The mapping diagram is a nice representation of the functions. It clearly shows the range, domain and codomain of a function taken into study. However, it comes in a detailed study of the function where two sets are made as domain and codomain, respectively, for a function. When the domain and codomain of a function match in several values, it is considered the set of real values that form the range of the function. The image or range of functions is a vast concept. The outputs of a function that has real values are also real numbers. The value we give to the function must produce a real-valued output after calculation.

Conclusion

Functions and Relations is a wide range in mathematics. In addition to the relevant topics introduced above, one should go through one-to-one function, many-to-one function, quadratic function, onto function and composition of functions, which include the function of a function. In conclusion, functions can be made very interesting and easy if learnt with dedication.