There are at least four basic arithmetic operations that can be performed on functions: addition, subtraction, multiplication, and division.
Basic Operations on Functions
Functions, much like numbers, can be combined using arithmetic operations. These operations create new functions based on the relationships between the original functions.
Addition
Adding two functions, f(x) and g(x), is denoted as (f + g)(x) and is defined as:
(f + g)(x) = f(x) + g(x)
Subtraction
Subtracting two functions, f(x) and g(x), is denoted as (f - g)(x) and is defined as:
(f - g)(x) = f(x) - g(x)
Multiplication
Multiplying two functions, f(x) and g(x), is denoted as (f * g)(x) and is defined as:
(f g)(x) = f(x) g(x)
Division
Dividing two functions, f(x) and g(x), is denoted as (f / g)(x) and is defined as:
(f / g)(x) = f(x) / g(x), where g(x) ≠ 0
It's crucial to note that the domain of the resulting function from division is restricted to values of x where the denominator function, g(x), is not equal to zero.
In addition to these four basic operations, function composition is another fundamental operation.
Function Composition
Function composition is applying one function to the result of another. It's denoted as (f ∘ g)(x) or f(g(x)) and defined as:
(f ∘ g)(x) = f(g(x))
This means you first evaluate g(x), and then use that result as the input for f(x). The domain of the composite function (f ∘ g)(x) consists of all x in the domain of g such that g(x) is in the domain of f.
Therefore, including composition, there are at least five common operations on functions.
