Function Composition

Definition

Function composition is the process of combining two or more functions to create a new function. The output of one function becomes the input of the next function, allowing for the creation of more complex mathematical relationships.

5 Must Know Facts For Your Next Test

1. Function composition is denoted by the symbol $\circ$, where $(f\circ g)(x) = f(g(x))$.
2. The domain of the composite function $f\circ g$ is the set of all $x$ values in the domain of $g$ for which $g(x)$ is in the domain of $f$.
3. Composing power functions, polynomial functions, rational functions, and inverse trigonometric functions can lead to new function types.
4. Inverse functions can be found by composing a function with its inverse, resulting in the identity function: $(f^{-1}\circ f)(x) = x$.
5. The graphs of composite functions can be obtained by transforming the graphs of the individual functions.

Review Questions

• Explain how function composition relates to the domain and range of functions.
  • The domain and range of the composite function $f\circ g$ are determined by the domains and ranges of the individual functions $f$ and $g$. The domain of $f\circ g$ 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 $f\circ g$ is the set of all $y$ values that can be obtained by applying $f$ to the range of $g$.
• Describe how function composition can be used to transform functions.
  • Function composition allows for the combination of various transformations, such as translations, reflections, and dilations, to create new functions. By composing functions, you can apply a sequence of transformations to the original function, leading to more complex and varied function types. This is particularly useful when working with power functions, polynomial functions, rational functions, and inverse trigonometric functions, as composing these functions can result in new function types.
• Analyze the relationship between function composition and inverse functions.
  • Function composition is closely tied to the concept of inverse functions. The composition of a function with its inverse results in the identity function, where $(f^{-1}\circ f)(x) = x$. This means that finding the inverse of a function can be achieved by composing the original function with its inverse. Additionally, function composition can be used to determine whether a function has an inverse, as the domain and range of the composite function must align for the inverse to exist.

Related terms

Domain and Range:

The domain of a function is the set of input values, while the range is the set of output values. Function composition requires considering the domains and ranges of the functions involved.

Transformation of Functions: Transformations, such as translations, reflections, and dilations, can be applied to functions, and these transformations can be combined through function composition.

Inverse Functions:

The inverse of a function undoes the original function, and function composition can be used to find inverse functions.