Addition of Two Real Functions: Mathematical Functions and Algebraic Operations

A real function is generally termed a real-valued function. It has a range that lies within the real numbers, that is non-root numbers and non-complex numbers.

• A real function is a factor that provides argument values.
• The formula P = f (x) denotes that the function ‘f’ assigns the value ‘P’ to the argument's value ‘x’.
• If the range of a function is a subset of real numbers, it is a real-valued function.
• If the domain of the function is a subset of real numbers, it is referred to as a real function.

Key Terms: Real Function, Functions, Real valued function, Real numbers, Non-root numbers, Non-complex numbers, Root numbers, Complex numbers

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Addition of Two Real Functions

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Let f : X → R and g : X → R be any two real functions, where X ⊂ R. Then, we define (f + g): X → R by

(f + g) (x) = f (x) + g (x), for all x ∈ X

Example: If f(x) = 3x² and g(x) = 2x + 3 are two real functions, then find (f + g)(x).

Ans. Given,

• f(x) = 3x²
• g(x) = 2x + 3

The addition of these two real functions can be given by

(f + g)(x) = f(x) + g(x) = 3x² + 2x + 3

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What is a Real-Valued Function?

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A real-valued function is one that has real numbers as its values. In other terms, it is a function that assigns a real number to each element in its domain.

• Calculus and, more generally, real analysis are primarily concerned with real-valued functions of a real variable (also known as real functions) and real-valued functions of several real variables.
• In particular, many function spaces are comprised of real-valued functions.

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Positive Real Functions 

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If a function in the form of F(x) falls into the 4 critical categories given below, then it is called a positive real function.

The 4 critical categories are:

1. For all real values of x, F(v) should have real values.
2. The polynomial F(v) must be a Hurwitz polynomial.
3. If we substitute v = jω, the real part of the function must be greater than or equal to zero when we divide the real and imaginary parts, which means it cannot be negative.

This is the most essential factor, and we often use it to eliminate concerns about whether or not a function is a positive real function.

4. F(x) must have simple poles when v = go is substituted, and the residues have to be real and positive. 

Critical Categories of Real Functions

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Properties of Positive Real Functions 

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Positive real functions have a number of significant characteristics, which are stated below: 

• F(v) should have Hurwitz polynomials in the numerator and denominator.
• The degree of the numerator of F(v) cannot be more than one greater than the degree of the denominator.
• To put it differently, (N-n) ought to be less than or equal to one.
• If F(v) is a positive real function, then its reverse has to be a positive real function as well.

Note that the sum of two or more positive real functions is also a positive real function, whereas the subtraction of two or more negative real functions is either a positive or a negative real function.

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Operations on Real Functions 

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In order to fully understand the critical aspects of real functions, we must concentrate on the methods below:

Adding Two Real Functions

After defining the functions j and k as j : Y → R and k : Y → R, respectively, two real functions such that Y is a subgroup of R, the sum of two real functions can be performed.

For any y ϵ  Y, (j + k) : Y → R can be defined as (j + k) (y) = j(y) + k(y).

Subtracting Two Real Functions

The difference between two real functions can be obtained by defining the functions j and k as j : Y → R and k : Y → R, respectively, as two real functions with Y as a subgroup of R.

For any y ϵ  Y, (j – k) : Y → R can be defined as (j – k) (y) = j(y) – k(y).

Multiplication of Real Functions

After defining the functions j and k as j : Y → R and k : Y → R, two real functions such that Y is a subgroup of R, the method of obtaining the product of two real examples of rational functions may be done.

For all y ϵ Y, jk : Y → R can be defined as (jk)(y) = j(y)k(y).

The quotient of Two Real Functions

After defining the functions j and k as j : Y → R and k : Y → R, the method of determining the quotient or division of two real functions can be undertaken. Y and R are two real functions that are subsets of one another.

For all y ϵ  Y, (j/k) : Y → R can be expressed as (j/k) (y) = j(y) / k(y).

Also Read:

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Things to Remember

• A real function has a range that lies within the real numbers, that is non-root numbers and non-complex numbers.
• If the range of a function is a subset of real numbers, it is a real-valued function.
• The addition of two real functions is given by (f + g) (x) = f (x) + g (x), for all x ∈ X.
• A real-valued function is one that has real numbers as its values.
• If F(v) is a positive real function, then its reverse has to be a positive real function as well.