Determine whether each pair of functions graphed are inverses.

Verified step by step guidance

1. Identify the given functions on the graph. The blue curve represents one function, and the orange curve represents another function.
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. This line serves as the mirror line for checking if the functions are inverses.
4. Check if the blue curve and the orange curve are symmetric with respect to the line y = x. This means that for every point (a, b) on the blue curve, there should be a corresponding point (b, a) on the orange curve, and vice versa.
5. Conclude whether the blue and orange curves are reflections of each other across the line y = x. If they are, then the functions are inverses; if not, they are not inverses.

Key Concepts

Inverse Functions

Inverse functions are functions that 'reverse' the effect of the original function. If a function f takes an input x and produces an output y, then its inverse f⁻¹ takes y as input and returns x. For two functions to be inverses, they must satisfy the condition 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 undo each other.

Function Composition

Function composition involves combining two functions to form a new function. For two functions f and g, the composition f(g(x)) applies g first and then f to the result. To verify if two functions are inverses, checking their compositions f(f⁻¹(x)) and f⁻¹(f(x)) is essential, as both should yield x for all x in the domain.