Algebra of Real Functions | Algebra - Mathematics PDF Download

Assume a case where Seema, whose monthly income is Rs. 15,000, spends Rs. 10,000. What will be her saving? Simple! Rs. 5,000. Saving = Income – Expenditure. Here, we see that the input and the output are the real numbers. We can say that real input gives a real output. Here, we will learn Real-valued functions and algebra of real functions. The above case is a representation of real mathematical functions and a case of subtraction in the 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.

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.
Note: It is also called pointwise multiplication.

Solved Example for You
Problem: Let f(x) = x³ and g(x) = 3x + 1 and a scalar, γ= 6. Find

1. (f + g) (x)
2. (f – g) (x)
3. (γf) (x)
4. (γg) (x)
5. (fg) (x)
6. (f / g) (x)

Solution: We have,

1. (f + g) (x) = f(x) + g(x) = x³ + 3x + 1.
2. (f – g) (x) = f(x) – g(x) = x³ – (3x + 1) = x³ – 3x – 1.
3. (γf) (x) = γ f(x) = 6x³ 
4. (γg) (x) = γ g(x) = 6 (3x + 1) = 18x + 6.
5. (fg) (x) = f(x) g(x) = x³ (3x +1) = 3x⁴ + x³.
6. (f / g) (x) = f(x) / g(x) = x³ / (3x +1), provided x ≠ – 1/3.