Algebra of Real Functions

The algebra of real functions refers to the study and application of algebraic operations on functions that map real numbers to real numbers.

A function can be thought of as a rule or set of rules which map an input to an output knows as its image. It is represented as x ⇢ Function ⇢ y. A real function refers to a function that maps real numbers to real numbers.

Algebra of real functions involves algebraic operations between two or more real functions. In this article we will learn in detail about algebra of real functions such addition, subtraction, multiplication and division between real function.

Algebra of Real Function

The algebra of real functions involves operations and properties that apply to functions map real numbers to real numbers. These operations include addition, subtraction, multiplication, division, composition, and inversion, among others.

Sometimes, while working on complex problems, combining two or more functions is required. In this case, one might need to combine a cube function with an absolute function.

Operation on Real Functions

We have explained in detail about different algebraic operation below.

Addition of Functions

Let f:X → R and g:X → R be any two real functions, where X ∈ R. Then we define (f + g): X → R by 

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

Let D(f) and D(g) be the domain of the functions "f" and "g," respectively. The domain in the case of function addition is given as

D(f + g) = D(f) ∩ D(g)

Example : Given f(x) = x + 3 and g(x) = 2x. Find (f + g)(x). 

Solution:

(f + g) (x) = f (x) + g (x).

Since f(x) = x + 3 and g(x) = 2x. 

(f + g)(x) = x + 3 + 2x = 3x + 3

Since domain for both the functions is real number R. The intersection of domain is also R. So, domain of (f + g)(x) is R. 

Subtraction of Functions

Let f:X → R and g:X → R be any two real functions, where X ∈ R. Then we define (f - g):X → R by

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

Let D(f) and D(g) be the domain of function "f" and "g" respectively. The domain in the case of function addition becomes.

D(f + g) = D(f) ∩ D(g)

Example: Given f(x) = 2x + 1 and g(x) = 3x. Find (f-g)(x). 

Solution:

(f - g) (x) = f (x) - g (x).

Since f(x) = 2x + 1 and g(x) = 3x. 

(f + g)(x) = 2x + 1 - 3x = 1 - x

Multiplication by a Scalar

Let f: X → R be a real function and "k" be any scalar belonging to R. Then the product kf is a function from X to R defined by 

(kf)(x) = kf(x), x ∈ X

The domain remains the same in this case. 

Multiplication of Two Real Functions

Let f : X → R and g : X → R be any two real functions, where X ⊆ R. Then product of these two functions i.e. f ∗ g : X → R is defined by

 (f × g) (x) = f (x) g (x) ∀ x ∈ X

Let D(f) and D(g) be the domain of function "f" and "g" respectively. In this case also the domain 

D(f × g) = D(f) ∩ D(g)

Example: Given f(x) = x² + 1 and g(x) = 1/x. Find (f × g)(x). 

Solution:

(f × g)(x) = f(x) g(x) = (x² + 1)(1/x) = \frac{x^{2} + 1}{x}. Domain remains the same as previous example in this case too. 

Quotient of Two Real Functions

Let f and g be two real functions defined from X → R. The quotient of f by g is denoted by \frac{f}{g} is a function defined from X → R as, 

(\frac{f}{g})(x) = \frac{f(x)}{g(x)}, provided g(x) ≠ 0.

Let D(f) and D(g) be the domain of function "f" and "g" respectively. In this case also the domain 

Domain for \frac{f}{g}: {x | x ∈ D_f ∩ D_g and g(x) ≠ 0}

Example: Given f(x) = x² + 1 and g(x) = (x + 1)/x. Find (\frac{f}{g})(x)

Solution:

Domain for f(x) is R and Domain for g(x) is R - {0}. Also, g(x) = 0 at x  = -1. 

\frac{f}{g}(x) = (x^{2} + 1)\frac{x + 1}{x} \\ = \frac{(x^{2} + 1)(x + 1)}{x}

We know, domain should be  {x | x ∈ D(f ) ∩ D(g) and g(x) ≠ 0 }. 

So, domain becomes R - {0,1}. 

Also Check

• Algebra of Derivative of Functions
• Domain and Range
• Relations and Functions

Example on Algebra of Real Functions

Example: Given the tables below: 

Calculate: 

• (f + g)(6)
• (f - g)(8)
• (f × g)(2)
• (f/g)(4)

Solution: 

• (f + g)(6) = f (6) + g(6) = 30 + (-14) = 16
• (f - g)(8) = f (8) - g(8) = 26 - (-22) = 48
• (f * g)(2) = f (2) x g(2) = 23 x 28 = 644
• (f / g)(4) = f (4) / g(4) = 16 / 32 = ½