Determine whether each pair of functions graphed are inverses.

Verified step by step guidance

1. Identify the two functions graphed: the orange curve and the blue curve.
2. Recall that two functions are inverses if their graphs are reflections of each other across the line y = x.
3. Observe the line y = x (green dashed line) on the graph.
4. Check if the orange curve and the blue curve are mirror images of each other across the line y = x.
5. If the orange curve and the blue curve are reflections of each other across the line y = x, then they are inverses; otherwise, they are not.

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.

Graphical Representation of Inverses

Graphically, two functions are inverses if their graphs are symmetric with respect to the line y = x. This means that if a point (a, b) lies on the graph of f, then the point (b, a) must lie on the graph of f⁻¹. This symmetry visually confirms that the functions reverse each other's operations.

Function Composition

Function composition involves combining two functions to form a new function. For two functions f and g, the composition f(g(x)) means applying g first and then applying f to the result. This concept is crucial for verifying if two functions are inverses, as it requires checking if the compositions yield the identity function.