Function Composition

Definition

Function composition is the process of combining two functions such that the output of one function becomes the input of another. This operation allows us to create new functions and analyze how different functions interact with each other. Understanding function composition is essential when exploring the properties of functions, such as injectivity, surjectivity, and bijectivity, as it reveals how these characteristics behave when functions are combined.

5 Must Know Facts For Your Next Test

1. Function composition is denoted as (f ∘ g)(x) = f(g(x)), where g is applied first and then f.
2. If f: A → B and g: B → C, then the composition f ∘ g is a function from A to C.
3. The order of composition matters; generally, f(g(x)) is not the same as g(f(x)).
4. For a composition of functions to be bijective, both component functions must also be bijective.
5. The composition of two injective functions is injective, and the composition of two surjective functions is surjective.

Review Questions

• How does function composition help in understanding properties like injectivity and surjectivity?
  • Function composition allows us to analyze how the properties of individual functions affect the resulting composite function. If we have two injective functions and we compose them, the resulting function will also be injective. Similarly, if we compose two surjective functions, we maintain surjectivity. This understanding is crucial for determining if a composite function retains specific characteristics based on its component functions.
• What conditions must be satisfied for the composition of two functions to be bijective?
  • For the composition of two functions to be bijective, both individual functions must be bijective themselves. This means each function must be injective and surjective. If either function lacks these properties, the composite function will not achieve a one-to-one correspondence between the elements of its domain and codomain, failing to be bijective.
• Analyze a scenario where you have an injective function followed by a surjective function. What can be said about the composition of these two functions?
  • When you have an injective function followed by a surjective function in a composition, the resulting composite function will be injective but not necessarily surjective. This occurs because while each input from the first function maps uniquely to an output, the second function covers all elements in its codomain, potentially allowing for multiple inputs to map to the same output in the composite function. Thus, while it retains uniqueness from the first part of the composition, it may lose complete coverage in terms of range.

Related terms

Injective Function: An injective function, or one-to-one function, is a function where every element in the codomain is mapped to by at most one element from the domain.

Surjective Function: A surjective function, or onto function, is a function where every element in the codomain has at least one element from the domain that maps to it.

Bijective Function: A bijective function is a function that is both injective and surjective, meaning there is a perfect one-to-one correspondence between elements of the domain and codomain.