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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) = x^{3 }and g(x) = 3x + 1 and a scalar, γ= 6. Find

- (f + g) (x)
- (f – g) (x)
- (γf) (x)
- (γg) (x)
- (fg) (x)
- (f / g) (x)

Solution: We have,

- (f + g) (x) = f(x) + g(x) = x
^{3 }+ 3x + 1. - (f – g) (x) = f(x) – g(x) = x
^{3 }– (3x + 1) = x^{3 }– 3x – 1. - (γf) (x) = γ f(x) = 6x
^{3 } - (γg) (x) = γ g(x) = 6 (3x + 1) = 18x + 6.
- (fg) (x) = f(x) g(x) = x
^{3 }(3x +1) = 3x^{4}+ x^{3}. - (f / g) (x) = f(x) / g(x) = x
^{3 }/ (3x +1), provided x ≠ – 1/3.

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