

How Real Functions Work: Key Features & Practice Problems
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:
F(v) should have real values for all real values of x.
F(v) must be a Hurwitz polynomial.
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.
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:
The numerator and denominator of F(v) must be Hurwitz polynomials.
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.
If F(v) is a positive real function, then the reciprocal of F(v) must also be a positive real function.
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.
FAQs on Real Functions Explained: Concepts, Types & Properties
1. What is a real function as per the Class 11 CBSE syllabus for 2025-26?
A real function, also known as a real-valued function, is a specific type of function where both its domain (the set of input values) and its codomain (the set of possible output values) are subsets of the set of real numbers (ℝ). In simpler terms, it's a rule that takes a real number as an input and produces another real number as an output. For a function f, this is denoted as f: X → Y, where X and Y are subsets of ℝ.
2. What is the difference between the domain, codomain, and range of a real function?
Understanding these three terms is crucial for working with real functions:
- Domain: This is the complete set of all possible input values (x-values) for which the function is defined. For example, in the function f(x) = 1/x, the domain includes all real numbers except 0.
- Codomain: This is the complete set of all possible output values that the function could theoretically produce. It is typically specified as the set of real numbers (ℝ) for real functions.
- Range: This is the actual set of output values (y-values) that the function produces from the given domain. The range is always a subset of the codomain. For f(x) = x², the domain is all real numbers, but the range is only non-negative real numbers [0, ∞).
3. How do you perform algebraic operations on two real functions?
If you have two real functions, f(x) and g(x), with a common domain, you can combine them using standard algebraic operations:
- Addition: (f + g)(x) = f(x) + g(x)
- Subtraction: (f - g)(x) = f(x) - g(x)
- Multiplication: (f * g)(x) = f(x) * g(x)
- Division: (f / g)(x) = f(x) / g(x), with the important condition that g(x) ≠ 0.
The domain of the resulting function is the intersection of the domains of f(x) and g(x), and for division, it excludes any x-values that make the denominator zero.
4. What are some common examples of real functions studied in mathematics?
Several types of real functions are fundamental in mathematics. Some key examples include:
- Polynomial Functions: Functions like f(x) = 3x² - 5x + 2, where the expression is a polynomial.
- Rational Functions: Functions that are a ratio of two polynomials, such as g(x) = (x + 1) / (x - 2).
- Modulus Function: The function f(x) = |x|, which gives the absolute value of x.
- Signum Function: This function indicates the sign of a number, returning -1 for negative numbers, 0 for zero, and 1 for positive numbers.
- Greatest Integer Function: f(x) = [x], which outputs the greatest integer less than or equal to x.
5. Why is finding the domain of a function the first and most important step?
Finding the domain is critical because it defines the very existence of the function. It tells you which input values are 'allowed' or will produce a valid, real number output. Attempting to use a value outside the domain can lead to mathematical errors, such as:
- Division by zero: In a rational function like f(x) = 1/(x-4), the value x=4 is not in the domain because it would cause division by zero.
- Taking the square root of a negative number: In a function like g(x) = √x, the domain is [0, ∞) because the square root of a negative number is not a real number.
By establishing the domain first, you ensure all subsequent calculations, graphing, and analysis are mathematically sound.
6. Is a relation like x = y² considered a real function? Why or why not?
No, the relation x = y² is not a function of x. A core rule of functions is that each input must map to exactly one unique output. In the case of x = y², if you choose an input like x = 9, the possible outputs for y are both 3 and -3 (since 3² = 9 and (-3)² = 9). Because one input (9) maps to two different outputs (3 and -3), it fails the 'vertical line test' and does not meet the definition of a function.
7. Can a constant, like f(x) = 5, be a real function?
Yes, f(x) = 5 is a perfect example of a constant function, which is a type of real function. For any real number you input for x, the output is always 5. Its domain is all real numbers (ℝ), and its range consists of a single value, {5}. It fully satisfies the definition of a function because every input maps to exactly one output, even though all inputs map to the same one.

















