Algebra of Functions
Algebra of functions deals with the arithmetic operations of the functions. For doing any arithmetic operation of two functions, their domains must be the same. Let us learn more about the algebra of functions along with formulas and examples.
1.  What is Algebra of Functions? 
2.  Algebra of Functions Formulas 
3.  Algebra of Functions Examples 
4.  Practice Questions on Algebra of Functions 
5.  FAQs on Algebra of Functions 
What is Algebra of Functions?
Algebra of functions means the operations of the functions, specifically the arithmetic operations. Algebra of functions mainly deals with the following four arithmetic operations of functions:
 Addition of functions
 Subtraction of functions
 Multiplication of functions
 Division of functions
Here are the formulas of all these operations.
Apart from these operations, we have another two important operations composite functions and inverse functions. To learn these, you cal click on the respective links. Let us study more about these formulas and solve a few examples also using the formulas.
Algebra of Functions Formulas
Any arithmetic operation (addition/subtraction/multiplication/division) of two functions is just nothing but the same operation of independent functions. Here are the formulas for each arithmetic operation for any two functions f(x) and g(x). Note that these formulas hold only when the domains of both the functions are the same (or restricted to the same domain).
Addition of Functions
The domain of the sum of two functions is the intersection of the domain of the independent functions. The sum of two functions at a given input is equal to the sum of the independent functions at the same input. i.e.,
 (f + g) (x) = f(x) + g(x)
Example: When f(x) = x^{2} + 2 and g(x) = x + 1, then
(f + g)(x) = f(x) + g(x)
= x^{2} + 2 + x + 1
= x^{2} + x + 3
Since the domain of each of f(x) and g(x) is the set of all real numbers, R, the domain of (f + g)(x) (which is R ∩ R = R) is R.
Subtraction of Functions
The domain of the difference of two functions is the intersection of the domain of the independent functions. The difference of two functions at a given input is equal to the difference of the independent functions at the same input. i.e.,
 (f  g) (x) = f(x)  g(x)
Example: When f(x) = x^{2} + 2 and g(x) = x + 1, then
(f  g)(x) = f(x)  g(x)
= x^{2} + 2  (x + 1)
= x^{2}  x + 1
Since the domain of each of f(x) and g(x) is the set of all real numbers, R, the domain of (f  g)(x) is R.
Multiplication of Functions
The domain of the product of two functions is the intersection of the domain of the independent functions. The product of two functions at a given input is equal to the product of the independent functions at the same input. i.e., The product of functions results in a binomial function, cubic function, or a polynomial function.
 (f · g) (x) = f(x) · g(x)
Example: When f(x) = x^{2} + 2 and g(x) = x + 1, then
(f · g)(x) = f(x) · g(x)
= (x^{2} + 2) · (x + 1)
= x^{3} + x^{2} + 2x + 2
Since the domain of each of f(x) and g(x) is the set of all real numbers, R, the domain of (f · g)(x) is R.
Division of Functions
The domain of the quotient of two functions is the intersection of the domain of the independent functions. But we have to take care of the extra condition, setting the denominator function to "not equal to 0" because if the denominator is 0, then the fraction is undefined. The quotient of two functions at a given input is equal to the quotient of the independent functions at the same input. i.e.,
 (f / g) (x) = f(x) / g(x), given g(x) ≠ 0
Example: When f(x) = x^{2} + 2 and g(x) = x + 1, then
(f / g)(x) = f(x) / g(x)
= (x^{2} + 2) / (x + 1)
Since the domain of each of f(x) is the set of all real numbers, R; and the domain of g(x) is the set of all real numbers except 1 (as x + 1 is in the denominator, x + 1 ≠ 0 ⇒ x ≠ 1). So the domain of (f / g)(x) is R  {1}.
Important Notes on Algebra of Functions:
For any two functions f(x) and g(x):
 (f + g) (x) = f(x) + g(x)
 (f  g) (x) = f(x)  g(x)
 (f · g) (x) = f(x) · g(x)
 (f / g) (x) = f(x) / g(x), g(x) ≠ 0
Related Topics:
Algebra of Functions Examples

Example 1: If f(x) = x + 2 and g(x) = x^{2}  3x + 2 then find (f + g)(x) and (f  g)(x) and find their domains.
Solution:
Using algebra of functions,
(f + g)(x) = f(x) + g(x)
= (x + 2) + (x^{2}  3x + 2)
= x^{2}  2x + 4(f  g)(x) = f(x)  g(x)
= (x + 2)  (x^{2}  3x + 2)
= x + 2  x^{2} + 3x  2
= x^{2} + 4xEach of f(x) and g(x) are polynomials and hence the domain of each of them is the set of all real numbers (R). So the domain of each of (f + g)(x) and (f  g)(x) is R ∩ R = R.
Answer: (f + g)(x) = x^{2}  2x + 4; (f  g)(x) = x^{2} + 4x; The domain of each of these is R.

Example 2: If f(x) = x + 2 and g(x) = x^{2}  3x + 2 then find (f g)(x) and (f / g)(x) and find their domains.
Solution:
Using algebra of functions,
(f g)(x) = f(x) g(x)
= (x + 2) (x^{2}  3x + 2)
= x^{3}  3x^{2} + 2x + 2x^{2}  6x + 4
= x^{3}  x^{2}  4x + 4(f / g)(x) = f(x) / g(x)
= (x + 2) / (x^{2}  3x + 2)
= (x + 2) / [(x  2) (x  1)]The domain of each of f(x) and g(x) is R. So the domain of (f g)(x) = R.
But for (f / g)(x), we have another condition that g(x) ≠ 0. So (x  2) (x  1) ≠ 0 ⇒ x ≠ 1 and x ≠ 2. So the domain of (f / g)(x) = R  {1, 2}.
Answer: (f g)(x) = x^{3}  x^{2}  4x + 4 and its domain is R; (f / g)(x) = (x + 2) / [(x  2) (x  1)] and its domain is R  {1, 2}.

Example 3: If f(x) = 1 / (x  2) and g(x) = √x  1, then find the domain of (f + g)(x) and (f / g)(x).
Solution:
f(x) has (x  2) in the denominator. So the domain of f(x) = {x  x ≠ 2}.
g(x) has x  1 inside the square root. So the domain of g(x) = {x  x ≥ 1}.
Domain of (f + g)(x):
The domain of (f + g) (x) = Domain of f(x) ∩ Domain of g(x)
= {x  x ≠ 2} ∩ {x  x ≥ 1}
= [1, 2) U (2, ∞)Domain of (f / g) (x):
The domain of (f / g)(x) is same as that of (f + g)(x) but we should take care of the extra condition g(x) ≠ 0.
When g(x) ≠ 0, its domain is {x  x > 1}.
So the domain of (f / g)(x)
= {x  x ≠ 2} ∩ {x  x > 1}
= (1, 2) U (2, ∞)Answer: The domain of (f + g)(x) = = [1, 2) U (2, ∞); The domain of (f / g)(x) = = (1, 2) U (2, ∞).
FAQs on Algebra of Functions
What is Algebra of Functions?
Algebra of functions talks about the addition, subtraction, multiplication, and division of functions. For any arithmetic operation of two functions at an input, we just have to apply the same operation for individual functions at the same input. For example, (f  g)(x) = f(x)  g(x).
What are Algebra of Functions Formulas?
The algebraic functions formulas are:
 (f + g) (x) = f(x) + g(x); (f  g) (x) = f(x)  g(x)
 (f g) (x) = f(x) g(x); (f / g) (x) = f(x) / g(x), g(x) ≠ 0
What are Algebra of Functions Examples?
If f(x) = x  7 and g(x) = x^{2}, then we have f(2) = 2  7 = 5 and g(2) = 2^{2} = 4. Then using algebra of functions:
 (f + g)(2) = f(2) + g(2) = 5 + 4 = 1.
 (f  g)(2) = f(2)  g(2) = 5  4 = 9.
 (f g)(2) = f(2) g(2) = (5) (4) = 20.
 (f / g)(2) = f(2) / g(2) = 5 / 4.
What is the Domain of Algebra of Functions?
The domain of sum, difference, and product of two functions is just the intersection of the domains of the individual functions. The domain of quotient of two functions is also the intersection of the domain of individual functions but we have to include the condition that the denominator function should not be 0.
What Are The Types of Functions for Algebra of Functions?
The types of functions which can be used for algebra of functions are one one function, onto function, many one function, constant functions. All the rules of algebra of functions apply to all of these functions.
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