Algebra of Real Functions

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 number input gives a real number 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.

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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).

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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

Question 1: 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)

Answer : 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.

Question 2: What is meant by functions in algebra?

Answer: A function refers to an equation that consists of only one answer for y for every x. A function assigns only one output to each input that is associated with a specified type. It is common that a function is named as g(x) or f(x) but not y.

Question 3: Explain what makes a function a function?

Answer: A relation from a set X to a set Y is known as a function in case each element of X has a relation to exactly one element in Y. For example, consider an element x in X, so there shall be only one element in Y that x can have a relation to.

Question 4: Is it possible for an equation to be a function?

Answer: An equation shall be considered a function only when for every x’s value there is only one corresponding value for y.

Question 5: When will a function be well defined?

Answer: A function will be well defined when it provides the same result when a change takes place in the representation of the input without the change taking place in the value of the input.