Algebra 2: Inverse Functions Flashcards

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.