Real-Valued Functions

Students who have studied Maths in class 12th and even 10th must be acquainted with the Real-Valued Functions as a large part of it was discussed in these classes. However, if not, then in this article, we will be explaining the Real-Valued Functions in detail, so make sure you do not skip any part. Real-valued functions are crucial while studying for IIT JEE Mains as a larger portion of the maths paper is asked from this particular topic. Therefore, there’s absolutely no escape!

Cracking IIT JEE Mains is a dream of many; however, only a fraction of the percentage of students are able to crack the exam and book their seats in government institutions. Each topic holds equal importance to excel in the exam, so do the Real-valued functions. So, if you have any questions or are facing difficulties regarding this topic, make sure you stick with us till the end. Without any further ado, let’s dive into it. 

What is the Real-Valued Function? 

The real-valued function, also known as the real function, is any function that lies within the real numbers. This means that the real-valued function is not the root numbers nor the complex numbers. It is an entity responsible for assigning argument values. P = F(x) is the notation which means for value x, a function given as F assigns the value given as P.

What are Polynomial Functions? 

Polynomial function can be defined as a function which either involves only positive integer exponents or the non-negative integer powers. 2x+5 is an example of the polynomial function. A polynomial function can also be stated through its degree. The degree of the polynomial function is a crucial part; it explains the function’s behaviour P(x), where x is probably enormous. 

General expression of polynomial of degree n in one variable is :

p(x) = an x^n + an-1 x^n-1 + ………… +a2 x^2 + a1 x + a0 , where an0

Examples of Polynomial functions 

As exponents, there are only two positive integers in a polynomial function. Different arithmetic operations can be performed for such functions, including subtraction, multiplication, addition, and division.

Some of the common examples of the polynomial functions are as follows: 

x^2 + 2x + 1 , 3x – 7 , 7x^3 + x^2 – 2

Types of Polynomial Functions 

Depending upon the degree of the polynomial, there are several types of polynomial functions, which are as follows: 

• Zero Polynomial Function : P(x) = a = ax0 
• Linear Polynomial Function: P(x) = ax + b
• Quadratic Polynomial Function: P(x) = ax^2 + bx + c 
• Cubic Polynomial Function: P(x) = ax^3 + bx^2 + cx + d 
• Quartic Polynomial Function: P(x) = ax^4 + bx^3 + cx^2 + dx + e

What are Linear Functions? 

A linear function is referred to as a function that includes one or more variables but no exponents. In the case of a linear function, if there are several variables in a function, then these variables need to be constant. 

The linear function generally forms a straight line whose degree is utmost  0 to 1. Apart from this, the linear functions are represented as linear algebra. The expression of the linear function graph is y = f(x) = px + q. It has both dependent and independent variables donated by y and x, respectively. Here p and q  are real numbers and q is called a constant term.

Characteristics of Linear Functions 

The following are the characteristics of linear functions. Let’s have a look- 

• Relation: It is termed as the group of the ordered pair
• Linear function: algebraic equation can be defined as if every term is the constant or the constant product along with a single variable
• Variable: It can be defined as a symbol used in maths to show quantity
• Steepness: It can be defined as the function’s rate deviated from the reference
• Direction: Horizontal, vertical, increasing, and decreasing

Graph of Linear Functions 

Three primary functions need to be followed religiously to form a linear function graph. These include: 

• The first step is to monitor the two points based on the equation y = px+q
• The next step involves plotting these graph points on the X-Y plane
• Using a ruler, join both points and form a straight line

Linear Function Table 

The below table showcases that the ordered pair notation is generalised in function and normal form. 

The values of x and y can be examined by looking at the table mentioned above. 

Considering the other table,

Looking at the table, it is evident the change rate is three between x and y, which can be written in the form of  y = x + 3

Conclusion 

Mentioned above is everything related to Real – valued functions and other important topics such as linear polynomials and others associated with the same. If you appear for the IIT JEE Mains, make sure you study every concept of Real – valued functions in detail and leave no gaps that might reflect in your result.