Composite and Inverse Functions Flashcards

Questions and Answers

What is the composition of functions?

Process of combining two functions where one function is performed first and the result is substituted in the other function.

What is the inverse of f(x)=2x+3?

f^-1(x)=(x-3)/2

What is the inverse of f(x)=x?

f^-1(x)=x

What are inverse functions?

f^-1(x) is said to be the inverse of f(x) if f(f^-1(x)) = f^-1(f(x)) = x. Signup and view all the answers

What is a function?

Relation where each element of the domain is paired with exactly one element of the range. Signup and view all the answers

What is a domain in the context of functions?

Set of all x-coordinates of the ordered pairs of that relation. Signup and view all the answers

What is a range of a function?

Set of all output values or the y-values of a function. Signup and view all the answers

What are the inverse points of the set {(1, -3), (-2, 3), (5, 1), (6, 4)}?

{ (-3, 1), (3, -2), (1, 5), (4, 6) } Signup and view all the answers

What are the inverse points of the set {(-5, 7), (-6, -8), (1, -2), (10, 3)}?

{ (7, -5), (-8, -6), (-2, 1), (3, 10) } Signup and view all the answers

Express the piecewise function f(x) = x if x > 0, = -x if x <= 0.

f(x) = { x for x > 0, -x for x <= 0 } Signup and view all the answers

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.