Algebra of Real Functions & Binary Operations | Mathematics (Maths) for JEE Main & Advanced PDF Download

ALGEBRA OF REAL FUNCTIONS
Real-valued Mathematical Functions
In mathematics, a real-valued function is a function whose values are real numbers. It is a function that maps a real number to each member of its domain. Also, we can say that a real-valued function is a function whose outputs are real numbers i.e., f: R → R (R stands for Real).

Algebra of Real Functions
In this section, we will get to know about addition, subtraction, multiplication, and division of real mathematical functions with another.

Addition of Two Real Functions
Let f and g be two real valued functions such that f: X → R and g: X → R where X ⊂ R. The addition of these two functions (f + g) : X → R  is defined by:

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

Subtraction of One Real Function from the Other
Let f : X → R and g : X → R be two real functions where X ⊂ R. The subtraction of these two functions (f – g): X → R  is defined by:

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

Multiplication by a Scalar
Let f: X → R be a real-valued function and γ be any scalar (real number). Then the product of a real function by a scalar γf: X → R is given by:
(γf) (x) = γ f(x), for all x ∈ X.

Equal functions 
Let f and g be two functions defined from A to B. Then f , g : A → B are equal if f (x) = g(x), x ∈ A .
If the function f and g are equal, then the subsets, graph of f and graph of g, of A x B are equal.

Q.1. 

find the value of k
Ans. We need to consider only one equation
2k = 6
k = 3

Q.2. Find the values of x and y.

Ans:
2x – 6 = 5
2x = 11
x = 5.5
4 – y = 3
y = 1

Multiplication of Two Real Functions
The product of two real functions say, f and g such that f: X → R and g: X → R, is given by

(fg) (x) = f(x) g(x), for all x ∈ X

Division of Two Real Functions
Let f and g be two real-valued functions such that f: X → R and g: X → R where X ⊂ R. The quotient of these two functions (f  ⁄ g): X → R  is defined by:

(f / g) (x) = f(x) / g(x), for all x ∈ X, but g(x) ≠ 0 for all x ∈ X

Note: It is also called point wise multiplication.

Solved Example
Q. Let f(x) = x³ and g(x) = 3x + 1 and a scalar, γ= 6. Find
(a) (f + g) (x)
(b) (f – g) (x)
(c) (γf) (x)
(d) (γg) (x)
(e) (fg) (x)
(f) (f / g) (x)
Sol: We have,
(a) (f + g) (x) = f(x) + g(x) = x³ + 3x + 1.
(b) (f – g) (x) = f(x) – g(x) = x³ – (3x + 1) = x³ – 3x – 1.
(c) (γf) (x) = γ f(x) = 6x³ 
(d) (γg) (x) = γ g(x) = 6 (3x + 1) = 18x + 6.
(e) (fg) (x) = f(x) g(x) = x³ (3x +1) = 3x⁴ + x³.
(f) (f / g) (x) = f(x) / g(x) = x³ / (3x + 1), provided x ≠ – 1/3.

BINARY OPERATIONS

We get a number when two numbers are either added or subtracted or multiplied or are divided. The binary operations associate any two elements of a set. The resultant of the two are in the same set. Binary operations on a set are calculations that combine two elements of the set (called operands) to produce another element of the same set.

The binary operations * on a non-empty set A are functions from A × A to A. The binary operation, *: A × A → A. It is an operation of two elements of the set whose domains and co-domain are in the same set.

Addition, subtraction, multiplication, division, exponential is some of the binary operations.

Properties of Binary Operation

• Closure property: An operation * on a non-empty set A has closure property, if a ∈ A, b ∈ A ⇒ a * b ∈ A.
• Additions are the binary operations on each of the sets of Natural numbers (N), Integer (Z), Rational numbers (Q), Real Numbers(R), Complex number(C).

The additions on the set of all irrational numbers are not the binary operations.

• Multiplication is a binary operation on each of the sets of Natural numbers (N), Integer (Z), Rational numbers (Q), Real Numbers(R), Complex number(C).

Multiplication on the set of all irrational numbers is not a binary operation.

• Subtraction is a binary operation on each of the sets of Integer (Z), Rational numbers (Q), Real Numbers(R), Complex number(C).

Subtraction is not a binary operation on the set of Natural numbers (N).

• A division is not a binary operation on the set of Natural numbers (N), integer (Z), Rational numbers (Q), Real Numbers(R), Complex number(C).
• Exponential operation (x, y) → x^y is a binary operation on the set of Natural numbers (N) and not on the set of Integers (Z).

Properties of Binary Operations
Commutative
A binary operation * on a set A is commutative if a * b = b * a, for all (a, b) ∈ A (non-empty set). Let addition be the operating binary operation for a = 8 and b = 9, a + b = 17 = b + a.

Associative
The associative property of binary operations hold if, for a non-empty set A, we can write (a * b) *c = a*(b * c). Suppose N be the set of natural numbers and multiplication be the binary operation. Let a = 4, b = 5 c = 6. We can write (a × b) × c = 120 = a × (b × c).

Distributive
Let * and o be two binary operations defined on a non-empty set A. The binary operations are distributive if a*(b o c) = (a * b) o (a * c) or (b o c)*a = (b * a) o (c * a). Consider * to be multiplication and o be subtraction. And a = 2, b = 5, c = 4. Then, a*(b o c) = a × (b − c) = 2 × (5 − 4) = 2. And (a * b) o (a * c) =  (a × b) − (a × c) = (2 × 5) − (2 × 4) = 10 − 6 = 2.

Identity
If A be the non-empty set and * be the binary operation on A. An element e is the identity element of a ∈ A, if a * e = a = e * a. If the binary operation is addition(+), e = 0 and for * is multiplication(×), e = 1.

Inverse
If a binary operation * on a set A which satisfies a * b = b * a = e, for all a, b ∈ A. a^-1 is invertible if for a * b = b * a= e, a-1 = b. 1 is invertible when * is multiplication.
Note: 1. Identity element is unique for whole given set A(if exists) i.e each element of given set A has same identity element.
2. But inverse element is different for different elements(if exists) , but unique.

Solved Example
Q.1. Show that division is not a binary operation in N nor subtraction in N.
Solution. Let a, b ∈ N
Case 1: Binary operation * = division(÷)
–: N × N→N given by (a, b) → (a/b) ∉ N (as 5/3 ∉ N)
Case 2: Binary operation * = Subtraction(−)
–: N × N→N given by (a, b)→ a − b ∉ N (as 3 − 2 = 1 ∈ N but 2−3 = −1 ∉ N).

Q.2. Let A = {−1, 1, 2, 3}. Construct the binary table corresponding to the binary operation “multiplication” on A.
Solution. We have the following binary (or composition) table: