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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(stands for Real).

Algebra of Real Functions | Algebra - Mathematics

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→and g: X→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→and g: X→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→be a real-valued function and γ be any scalar (real number). Then the product of a real function by a scalar γf: X→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→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→and g: X→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) = xand 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) = x+ 3x + 1.
  2. (f – g) (x) = f(x) – g(x) = x– (3x + 1) = x– 3x – 1.
  3. (γf) (x) = γ f(x) = 6x
  4. (γg) (x) = γ g(x) = 6 (3x + 1) = 18x + 6.
  5. (fg) (x) = f(x) g(x) = x(3x +1) = 3x4 + x3.
  6. (f / g) (x) = f(x) / g(x) = x/ (3x +1), provided x ≠ – 1/3.
The document Algebra of Real Functions | Algebra - Mathematics is a part of the Mathematics Course Algebra.
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FAQs on Algebra of Real Functions - Algebra - Mathematics

1. What is the algebra of real functions?
Ans. The algebra of real functions is a branch of mathematics that deals with the operations and properties of functions defined on the real numbers. It involves operations such as addition, subtraction, multiplication, and division of functions, as well as the study of their properties, compositions, and inverses.
2. What are the basic operations in the algebra of real functions?
Ans. The basic operations in the algebra of real functions are addition, subtraction, multiplication, and division. Addition of two functions f(x) and g(x) is defined as (f+g)(x) = f(x) + g(x), subtraction is defined as (f-g)(x) = f(x) - g(x), multiplication is defined as (f*g)(x) = f(x) * g(x), and division is defined as (f/g)(x) = f(x) / g(x), where g(x) ≠ 0.
3. How do you find the composition of two real functions?
Ans. To find the composition of two real functions f(x) and g(x), denoted as (f∘g)(x), we substitute the expression of g(x) into f(x). In other words, we replace every occurrence of the variable x in the function f(x) with the function g(x). The resulting composite function (f∘g)(x) represents the composition of f(x) and g(x).
4. What is the inverse of a real function?
Ans. The inverse of a real function f(x) is denoted as f^(-1)(x) and represents a function that "undoes" the original function. In other words, if f(x) maps an input x to an output y, then f^(-1)(x) maps the output y back to the original input x. The inverse function is obtained by swapping the input and output variables in the original function and solving for the output variable.
5. What are some properties of the algebra of real functions?
Ans. Some properties of the algebra of real functions include commutativity, associativity, distributivity, and the existence of an identity element. Addition and multiplication of real functions are commutative and associative operations. Distributivity holds for the multiplication of functions over addition or subtraction. Additionally, the constant function 0(x) = 0 serves as the additive identity, and the constant function 1(x) = 1 serves as the multiplicative identity.
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