Determine whether each pair of functions graphed are inverses.

Key Concepts

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other. If a function f takes an input x and produces an output y, then its inverse f⁻¹ takes y back to x. For two functions to be inverses, the composition of the functions must yield the identity function, meaning f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all x in their domains.

Graphing Functions and Inverses

When graphing functions and their inverses, a key characteristic is that the graphs of inverse functions are symmetric with respect to the line y = x. This means that if a point (a, b) lies on the graph of a function, then the point (b, a) will lie on the graph of its inverse. This symmetry is crucial for visually determining if two functions are inverses.

The Line y = x

The line y = x serves as a reference line for identifying inverse functions graphically. Any point on this line has equal x and y coordinates, which means that for a function and its inverse, their graphs will intersect this line at points where they are equal. This line helps in verifying the symmetry of the graphs of two functions to determine if they are inverses.