Algebra of Real Functions & Binary Operations - (Maths) for JEE Main

ALGEBRA OF REAL FUNCTIONS

Real-valued mathematical functions
A real-valued function is a function whose outputs are real numbers. If the domain is a subset X of the real numbers, a real-valued function is written as f : X → R. For each x ∈ X the function assigns a real number f(x).

Algebra of real functions - basic operations

If f and g are real-valued functions defined on the same domain X ⊂ R, we can combine them by simple algebraic rules to obtain new functions. These operations are defined pointwise: the value of the combined function at x ∈ X is obtained by performing the corresponding arithmetic on f(x) and g(x).

Addition

If f : X → R and g : X → R, the sum function f + g : X → R is defined by

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

Subtraction

If f : X → R and g : X → R, the difference function f - g : X → R is defined by

(f - g)(x) = f(x) - g(x), for all x ∈ X.

Scalar multiplication

Let f : X → R and let γ be a real number (scalar). The scalar multiple γf : X → R is defined by

(γf)(x) = γ·f(x), for all x ∈ X.

Pointwise multiplication

If f : X → R and g : X → R, the product function fg : X → R is defined by

(fg)(x) = f(x)·g(x), for all x ∈ X.

Pointwise division (quotient)

If f : X → R and g : X → R, the quotient (when defined) f/g : X → R is given by

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

Equal functions and graphs

Two functions f, g : A → B are equal if and only if f(x) = g(x) for every x ∈ A. If f and g are equal then their graphs (as subsets of A × B) are the same.

Example questions and solutions

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 = 11/2 = 5.5

4 - y = 3

y = 1

Solved example - combining standard functions

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:

(f + g)(x) = f(x) + g(x) = x³ + 3x + 1.

(f - g)(x) = f(x) - g(x) = x³ - (3x + 1) = x³ - 3x - 1.

(γf)(x) = γ·f(x) = 6x³.

(γg)(x) = γ·g(x) = 6(3x + 1) = 18x + 6.

(fg)(x) = f(x)·g(x) = x³(3x + 1) = 3x⁴ + x³.

(f / g)(x) = f(x)/g(x) = x³/(3x + 1), provided 3x + 1 ≠ 0, i.e. x ≠ -1/3.

BINARY OPERATIONS

A binary operation on a non-empty set A is a rule that combines any two elements of A to produce another element of A. Formally, a binary operation * on A is a function * : A × A → A. The inputs are ordered pairs (a, b) with a, b ∈ A, and the output a * b must lie in A.

Common examples

• Addition is a binary operation on the sets N (natural numbers), Z (integers), Q (rationals), R (reals) and C (complex numbers).
• Addition is not a binary operation on the set of all irrational numbers because the sum of two irrationals can be rational.
• Multiplication is a binary operation on N, Z, Q, R and C.
• Multiplication is not a binary operation on the set of all irrational numbers because the product of two irrationals can be rational.
• Subtraction is a binary operation on Z, Q, R and C, but not on N.
• Division is not a binary operation on N, Z, Q, R or C because division may not produce an element of the same set (and division by zero is undefined).
• The exponentiation map (x, y) → x^y is a binary operation on N (when y is a natural number) but is not a binary operation on Z (exponent may not be an integer-preserving operation for negative bases and integer exponents).

Properties of binary operations

Closure

Closure means: for all a, b ∈ A, the element a * b ∈ A. Closure is part of the definition of a binary operation.

Commutativity

A binary operation * on A is commutative if a * b = b * a for all a, b ∈ A. Example: addition and multiplication of real numbers are commutative.

Associativity

A binary operation * on A is associative if (a * b) * c = a * (b * c) for all a, b, c ∈ A. Example: addition and multiplication of real numbers are associative.

Distributivity

Let * and o be two binary operations on A. We say * is left-distributive over o if a * (b o c) = (a * b) o (a * c) for all a, b, c ∈ A. Similarly, * is right-distributive over o if (b o c) * a = (b * a) o (c * a) for all a, b, c ∈ A. Example: multiplication is distributive over addition in R.

Identity element

An element e ∈ A is an identity element for * if a * e = a = e * a for every a ∈ A. If an identity exists it is unique. Example: 0 is the additive identity in R; 1 is the multiplicative identity in R.

Inverse element

For a binary operation * with identity e, an element b ∈ A is an inverse of a ∈ A if a * b = e = b * a. If an inverse exists it is unique for the given a. Example: for multiplication on R \ {0}, the inverse of a is 1/a.

Note:

• Identity element (if it exists) is the same for every element of A. It is unique.
• Inverse of a particular element (if it exists) may differ from inverses of other elements, but for a given element the inverse is unique.

Worked examples - binary operations

Q.1. Show that division is not a binary operation in N nor subtraction in N.

Solution. 

Let a, b ∈ N.

Consider the operation division (÷): N × N → N given by (a, b) → a / b. For a = 5 and b = 3, a / b = 5/3 which is not a natural number. Therefore division is not a binary operation on N.

Consider subtraction (-): N × N → N given by (a, b) → a - b. For a = 2 and b = 3, a - b = -1 which is not a natural number. Therefore subtraction is not a binary operation on N.

Q.2. Let A = {-1, 1, 2, 3}. Construct the binary table corresponding to the binary operation "multiplication" on A.

Solution.

We list the products of all ordered pairs from A. The binary (multiplication) table has rows and columns indexed by -1, 1, 2, 3 and each cell is the product of the corresponding row and column elements.

Remarks and applications

Working with algebra of real functions is fundamental for calculus, modelling, and problem solving. Binary operations and their properties are basic building blocks for algebraic structures such as groups, rings and fields which appear in higher mathematics and in competition problems. Recognising where operations are defined, where identities and inverses exist, and which properties (commutativity, associativity, distributivity) hold is essential for manipulating expressions correctly.