Real Functions

From the cartesian point of view, here, X is a function of Y because the elements of X are directly related to the elements of Y. Here, 1 directly maps with D; 2 and 3 are directly related to C. As a result of this, we can understand that a function is a process that connects each element of set X to a single element of a set Y. The process for reading this is Y= f(x). These are the simplest operations of function. Next, we will look into what is a real function?

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What is a Real Function?

A function whose range lies within the real numbers i.e., non-root numbers and non-complex numbers, is said to be a real function, also called a real-valued function.

A real function is an entity that assigns values to arguments. The notation P = f (x) means that to the value x of the argument, the function f assigns the value P. Sometimes, we also use the notation f: x ↦ P, in words, the function f sends x to P. The most usual way of specifying this assignment is by some formula, that is, the function value P can be obtained by substituting x to a specific formula that identifies the given function.

Any function in the form of F(x) is called a positive real function, if it falls under these four critical categories:

1. F(v) should have real values for all real values of x.

2. F(v) must be a Hurwitz polynomial.

3. If we substitute v = j*ω then on splitting up the real and imaginary parts, the real part of the function must be more than or equivalent to zero, which means it should not be negative. This is the most critical condition, and we frequently use this theory to clear doubts regarding the fact that the function is a positive real function or not.

4. On substituting v = go, F(x) should own simple poles, and the residues must be real and positive.

Properties of Positive Real Function

There are some important properties of a positive real function, which are listed below:

1. The numerator and denominator of F(v) must be Hurwitz polynomials.

2. The degree of the numerator of F(v) must not be more than the degree of the denominator by more than 1. In other words, (N-n) must be lesser than or equal to one.

3. If F(v) is a positive real function, then the reciprocal of F(v) must also be a positive real function.

4. Do not forget that the addition of two or more positive real functions is also a positive real function, but in the case of the subtraction, either it will be a positive real function or a negative real function.

Operations on Real Functions

Now, we have to pay attention to the following procedures in order to understand the basic problems of real functions.

Adding Two Real Functions: The process of summation of two real functions can be done after defining the functions j and k as j: Y ⟶R and k: Y ⟶R is two real functions, such that Y is a subset of R. Then (j + k): Y ⟶R can be defined as (j + k)(y) = j(y) + k(y), for all y ϵ Y.

Subtracting Two Real Functions: The process of finding out the difference of two real functions can be done after defining the functions j and k as j: Y ⟶R and k: Y ⟶R are two real functions, such that Y is a subset of R. Then (j - k): Y ⟶R can be defined as (j – k)(y) = j(y) – k(y), for all y ϵ Y.

Multiplication of Real Function: The process of finding out the product of two real-life examples of rational functions can be done after defining the functions j and k as j: Y ⟶R and k: Y ⟶R are two real functions, such that Y is a subset of R. Then jk: Y ⟶R can be defined as (jk)(y) = j(y)k(y), for all y ϵ Y.

The quotient of Two Real Functions: The process of finding out the quotient or division of two real functions can be done after defining the functions j and k as j: Y ⟶R and k: Y ⟶R are two real functions, such that Y is a subset of R. Then (j/k): Y ⟶R can be defined as (j/k)(y)=j(y) / k(y), for all y ϵ Y.

Solved Example

If function ‘h’ is defined by

l(x) = 3x2 - 7x - 5,

find l(x - 2).

Solution: 

By the theory and concept of function,

Substitute x by x -2 in the formula of function written below,

l(x - 2) = 3 (x - 2)2 - 7 (x - 2) - 5

Expand and group the like terms for your convenience. For expansions, use the basic algebraic theorems on polynomial multiplications and additions. Do not forget to look upon the degree of the polynomials for the accuracy of results. 

l (x - 2) = 3 ( x² - 4 x + 4 ) - 7 x + 14 - 5

After the expansion and grouping of like terms, our job is to simplify the terms and make a compact polynomial after making the required summations and subtractions.

= 3 x² - 19 x + 7.