How can a graph help show the relationship between a function and its inverse?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

The graph of a function and its inverse are related through geometric symmetry. Specifically, the graph of an inverse function is a reflection of the original function's graph across the line $y = x$ .

This relationship occurs because an inverse function effectively swaps the roles of the independent and dependent variables. If a point $(a, b)$ lies on the graph of the original function, the point $(b, a)$ must lie on the graph of the inverse function. By reflecting the original curve across the diagonal line $y = x$ , you visualize how the inverse "undoes" the mapping of the original function, mapping the output back to the original input.1