The composition of two functions, denoted as f ∘ g, is a new function that is created by applying one function (g) to the input of another function (f). The resulting function represents the combined effect of applying both functions in succession.
5 Must Know Facts For Your Next Test
The composition of functions, f ∘ g, is defined as (f ∘ g)(x) = f(g(x)), where x is the input variable.
The domain of the composite function f ∘ g is the set of all x values in the domain of g for which g(x) is in the domain of f.
The range of the composite function f ∘ g is the set of all f(g(x)) values, where x is in the domain of g.
Composition of functions is not commutative, meaning that in general, f ∘ g ≠ g ∘ f.
Composition of functions is associative, meaning that (f ∘ g) ∘ h = f ∘ (g ∘ h).
Review Questions
Explain the process of finding the composition of two functions, f(x) and g(x).
To find the composition of two functions, f(x) and g(x), you first apply the function g(x) to the input variable x, and then apply the function f(x) to the result of g(x). The composition is denoted as (f ∘ g)(x) = f(g(x)). This means that the input variable x is first substituted into the function g(x), and the resulting output is then substituted into the function f(x) to obtain the final output value.
Describe the relationship between the domains and ranges of the composite function f ∘ g and the original functions f and g.
The domain of the composite function f ∘ g is the set of all x values in the domain of g for which g(x) is in the domain of f. The range of the composite function f ∘ g is the set of all f(g(x)) values, where x is in the domain of g. The domain and range of the composite function are determined by the interplay between the domains and ranges of the original functions f and g, as the output of g must be a valid input for f in order for the composition to be defined.
Explain why the composition of functions is not commutative, but is associative.
The composition of functions is not commutative, meaning that in general, f ∘ g ≠ g ∘ f. This is because the order in which the functions are applied matters, as the output of one function becomes the input for the other function. However, the composition of functions is associative, meaning that (f ∘ g) ∘ h = f ∘ (g ∘ h). This property allows for the functions to be applied in any order without changing the final result, as long as the composition is well-defined.