Algebra 2: Inverse Functions Flashcards

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Questions and Answers

What is a relation?

a set of pairs of input and output values

What is the domain of a relation?

the set of all input values of a relation

What is the range of a relation?

the set of all output values of a relation

What is a function?

A relationship from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range. Signup and view all the answers

What is the vertical line test?

A test used to determine whether a relation is a function by checking if a vertical line touches 2 or more points on the graph of a relation Signup and view all the answers

What is a one-to-one function?

A property of functions where the same value for y is never paired with two different values of x. Signup and view all the answers

What is the horizontal line test?

If any horizontal line only crosses a graph at most once, it has an inverse function. Signup and view all the answers

What is the composition of functions?

The process of combining two functions together. Signup and view all the answers

What are inverse functions?

Two functions f and g are inverse functions if and only if both of their compositions are the identity function. Signup and view all the answers

What does the domain of a function become for its inverse?

the range Signup and view all the answers

What does the range of a function become for its inverse?

the domain Signup and view all the answers

Do the coordinates (-3, 8) and (8, -3) represent inverse coordinates?

True Signup and view all the answers

Do the coordinates (-2, 5) and (-5, 2) represent inverse coordinates?

False Signup and view all the answers

What is the inverse of the coordinate (2, -7)?

(-7, 2) Signup and view all the answers

What are the steps to solve algebraically for the inverse of a function?

<ol> <li>Replace f(x) with y in the equation for f(x). 2. Interchange x and y. 3. Solve for y. 4. Replace y with f⁻¹(x).</li> </ol> Signup and view all the answers

What is the inverse function, given f(x) = 2x - 16?

f⁻¹(x) = ½x + 8 Signup and view all the answers

What is the inverse of the equation y = x/6?

f⁻¹(x) = 6x Signup and view all the answers

What is the inverse of x squared?

square root of x Signup and view all the answers

Study Notes

Definitions and Concepts

Relation: A set comprising pairs of input and output values.
Domain: The complete set of all input values for a relation.
Range: The complete set of all output values for a relation.
Function: A specific type of relation that assigns each element from the domain to exactly one element in the range. The vertical line test is used to determine if a relation qualifies as a function.
Vertical Line Test: Determines if a relation is a function by observing if a vertical line intersects the graph at no more than one point.
One-to-One Function: A function type where no two different x-values correspond to the same y-value. It passes the horizontal line test.
Horizontal Line Test: If any horizontal line intersects a graph at most once, the function has an inverse.

Function Composition and Inverses

Composition of Functions: The action of combining two functions. Given f(x) and g(x), the composite function f(g(x)) means g is the input for f.
Inverse Functions: Functions f and g are considered inverses if their compositions yield the identity function; that is, f(g(x)) = x and g(f(x)) = x. Their graphs reflect over the line y = x, and for ordered pairs (x, y), the inverse is (y, x). Notation for inverse function is f⁻¹(x).

Domain and Range in Inverses

The domain of a function becomes the range in its inverse.
The range of a function transforms into the domain in its inverse.

Inverse Coordinate Pairs

Coordinates (-3, 8) and (8, -3) are inverse pairs.
Coordinates (-2, 5) and (-5, 2) do not represent inverse pairs.
The inverse of the coordinate (2, -7) is (-7, 2).

Steps for Finding Inverses Algebraically

Replace f(x) with y in the function’s equation.
Switch x and y.
Solve for y.
Substitute y back with f⁻¹(x).

Examples of Inverse Functions

Inverse of f(x) = 2x - 16 is f⁻¹(x) = ½x + 8.
For the function y = x/6, the inverse is f⁻¹(x) = 6x.
The inverse of x squared is √x.
Visual confirmation can be obtained by graphing both functions and the line y = x, demonstrating reflection.

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Description

Test your knowledge of inverse functions in Algebra 2 with these flashcards. Each card presents key terms and definitions to help reinforce your understanding of relations, domains, and ranges. Perfect for quick reviews and study sessions.