Composite and Inverse Functions Flashcards

Study Notes

Functions and Their Properties

Composition of Functions involves combining two functions where the output of the first function becomes the input for the second function.
A function is a relation that pairs each element of the domain precisely with one element of the range.

Inverse Functions

The inverse of the function f(x) = 2x + 3 is f^(-1)(x) = (x - 3) / 2.
The inverse of the function f(x) = x is f^(-1)(x) = x, indicating that it is its own inverse.
For a function f(x), the inverse function f^(-1)(x) is verified if f(f^(-1)(x)) = f^(-1)(f(x)) = x, achieved by swapping input and output variables in the function.

Domains and Ranges

The domain of a function is defined as the complete set of possible x-coordinates from its ordered pairs.
The range refers to all possible output values (y-values) that can result from the function’s inputs.

Finding Inverses of Point Sets

For the set {(1, -3), (-2, 3), (5, 1), (6, 4)}, the inverses are calculated as {(-3, 1), (3, -2), (1, 5), (4, 6)} by swapping each pair.
For the set {(-5, 7), (-6, -8), (1, -2), (10, 3)}, the inverses obtained are {(7, -5), (-8, -6), (-2, 1), (3, 10)} by interchanging the coordinates.

Piecewise Functions

The piecewise function is defined by:
- f(x) = x for x > 0
- f(x) = -x for x ≤ 0
This definition demonstrates how the function's behavior varies depending on the value of x.

Description

Test your understanding of composite and inverse functions with these flashcards. Each card presents an important concept or definition related to the topic, making it a great tool for students in mathematics. Enhance your learning with key examples and definitions to solidify your knowledge.