Follow the Inner Function before the Outer One
In f(g(x)), g acts on the supplied input before f acts on its result.
About 25 minutes, plus any extended practice
Open lesson 1Precalculus: Composition and Inverse Functions
Function operations must respect input order and domain. A composition feeds one output into another function, while an inverse reverses a one-to-one relationship and exchanges the original domain and range.
Before you begin
- Evaluate functions; solve linear equations; interpret square-root domains and restricted quadratic inputs.
Choose by skill rather than age alone. Read with an adult when useful, revisit a lesson, or explore another grade.
Follow the lesson sequence
Require the Inner Output to Fit the Outer Domain
A composed function exists only when its inner function is defined and that output belongs to the outer function’s domain.
About 25 minutes, plus any extended practice
Open lesson 2Reverse a Linear Function’s Operations
A nonconstant linear function is one-to-one.
About 25 minutes, plus any extended practice
Open lesson 3Restrict a Quadratic to Make an Inverse Function
Squaring on all real inputs loses sign information and is not one-to-one.
About 25 minutes, plus any extended practice
Open lesson 4Review and transfer
After the lessons, try 4 new review cases and a practical task. Each answer has an explanation and a link back to the related lesson.
Open the unit reviewGuest answers are temporary. Keep your own paper or document; this unit does not certify completion or record a school grade.
Topic references
- OpenStax mathematical topic reference: topic reference — Precalculus: Composition and Inverse Functions
These references informed the topic selection. They do not certify complete standards alignment.