Combining functions involves performing mathematical operations on two or more functions to create a new function. This can be done through algebraic operations or function composition.
Algebraic Operations on Functions
You can combine functions using basic algebraic operations: addition, subtraction, multiplication, and division. Let's say you have two functions, f(x) and g(x).
- Addition: (f + g)(x) = f(x) + g(x)
- Subtraction: (f - g)(x) = f(x) - g(x)
- Multiplication: (f g)(x) = f(x) g(x)
- Division: (f / g)(x) = f(x) / g(x), provided g(x) ≠ 0
Example:
Let f(x) = x2 and g(x) = 2x + 1
- (f + g)(x) = x2 + 2x + 1
- (f - g)(x) = x2 - (2x + 1) = x2 - 2x - 1
- (f g)(x) = x2 (2x + 1) = 2x3 + x2
- (f / g)(x) = x2 / (2x + 1), where x ≠ -1/2
Function Composition
Another way to combine functions is through function composition. This means applying one function to the result of another. The composition of f with g, denoted f(g(x)) or (f ∘ g)(x), means you first evaluate g(x), and then you use that result as the input for f(x).
Example:
Let f(x) = x + 2 and g(x) = 3x
Then:
- f(g(x)) = f(3x) = (3x) + 2 = 3x + 2
- g(f(x)) = g(x + 2) = 3(x + 2) = 3x + 6
Important Note: Function composition is not commutative in general, meaning f(g(x)) is not always equal to g(f(x)).
Finding the Domain of Combined Functions
When combining functions, you also need to consider the domain of the resulting function.
- For algebraic operations (addition, subtraction, multiplication, and division), the domain of the combined function is the intersection of the domains of the original functions, excluding any values that make the denominator zero in the case of division.
- For function composition f(g(x)), you must ensure that x is in the domain of g, and g(x) is in the domain of f.
Summary
Finding the combination of a function involves either performing algebraic operations or function composition. Algebraic operations combine function outputs, while function composition uses the output of one function as the input for another. Always consider the domain of the resulting function.
