Basic operations on functions
We can define the four basic operations--sum, difference, product, and quotient--as follows:
If the ranges of functions f and g are subsets of the real numbers, then
1. the sum of f and g, denoted as f+g, is defined as
(f+g)(x)= f(x) + g(x)
2. the difference of f and g, denoted as f-g, is defined as
(f-g)(x)= f(x) - g(x)
3. the product of f and g, denoted as f.g, is defined as
(f . g)(x) =f(x) . g(x)
4. the quotient of f and g, denoted as f/g, is defined as
(f/g)(x) = f(x)/g(x)
Did you realize that the basic operations on functions
are so simple and straightforward?
Example:
If f(x) = 2x + 1 and g(x) = x - 3, find (f + g)(x), (f - g)(x) and their domains.
Solution:
By using the above operations, we can write
- (f + g)(x) = f(x) + g(x) = (2x + 1) + (x - 3) = 3x - 2
The domain of (f + g)(x) is the set of real numbers since the domains of f and g are the set of real
numbers.
- (f - g)(x) = f(x) - g(x) = (2x + 1) - (x - 3) = x + 4
The domain of (f - g)(x) is also the set of real numbers since the domains of f and g are the set of real numbers.
Example:
If f(x) = x2 - 1, g(x) = x +1, find (f . g)(x) and (f/g)(x)
Solution:
- (f . g)(x) = f(x) . g(x) = (x2 - 1)(x+1) =x3 + x2 - x - 1
- (f/g)(x) = f(x)/g(x) = (x2 - 1)/(x + 1) = (x +1)(x -1)/(x+1) = (x - 1)