Composition of functions

Definition

The composition of functions is a mathematical operation that takes two functions, say f and g, and combines them into a new function, denoted as (f ∘ g)(x) = f(g(x)). This means that the output of the function g is fed as the input to the function f. Understanding this concept is crucial, as it not only emphasizes how functions can be combined to create new outputs, but it also highlights the relationships between different functions and their properties.

5 Must Know Facts For Your Next Test

1. Composition of functions is not always commutative, meaning that f(g(x)) does not necessarily equal g(f(x)).
2. The domain of the composition of functions must be taken from the domain of the inner function; if g(x) is not in the domain of f, then (f ∘ g)(x) is undefined.
3. When composing functions, it is often useful to evaluate the inner function first before applying the outer function.
4. If f and g are both defined on the same domain, their composition (f ∘ g) will have a domain based on g's range.
5. Understanding the composition of functions helps in solving complex equations by breaking them down into simpler parts.

Review Questions

• How does the composition of functions demonstrate the relationship between different functions in a mathematical context?
  • The composition of functions shows how one function can operate on the output of another, illustrating interdependencies between functions. For instance, if you have two functions f and g, composing them into (f ∘ g)(x) means that you first evaluate g at x and then apply f to that result. This highlights how outputs from one function can serve as inputs for another, thereby showing connections in their behavior and transformations.
• What conditions must be satisfied for the composition of two functions to be defined, and why are these conditions important?
  • For the composition of two functions to be defined, the output of the inner function must fall within the domain of the outer function. This condition is important because if g(x) produces an output that f cannot accept as an input, then (f ∘ g)(x) becomes undefined. Therefore, understanding these domains helps prevent errors in calculations and ensures that compositions are valid operations.
• Evaluate and analyze how changes in one function affect the overall behavior of their composition when considering specific examples.
  • Analyzing how changes in one function affect their composition can be observed through examples like f(x) = x + 2 and g(x) = 3x. If we compose these functions as (f ∘ g)(x), we find that (f ∘ g)(x) = 3x + 2. If we alter g to g(x) = x^2 instead, now (f ∘ g)(x) results in x^2 + 2. This demonstrates how changing g affects not just its own output but also modifies the entire behavior and characteristics of the composed function, illustrating interdependencies that can lead to different graphs or behaviors based on slight adjustments.

Related terms

Function: A relation between a set of inputs and a set of possible outputs, where each input is related to exactly one output.

Inverse Function: A function that reverses the effect of another function, such that if f(x) = y, then the inverse function f^{-1}(y) = x.

Identity Function: A function that always returns the same value that was used as its input, usually denoted as f(x) = x.