This tutorial explains the most important properties of inverse functions, with examples and graphical interpretations. For a general introduction, see the inverse functions tutorial.
Only one-to-one functions have inverse functions. If \( g \) is the inverse of \( f \), then \( f \) is the inverse of \( g \), and both functions must be one-to-one.
Functions \( f \) and \( g \) are inverses of each other if and only if: \[ (f \circ g)(x) = x, \quad x \text{ in the domain of } g \] and \[ (g \circ f)(x) = x, \quad x \text{ in the domain of } f \]
Let \[ f(x) = 3x \quad \text{and} \quad g(x) = \frac{x}{3}. \]
Then: \[ (f \circ g)(x) = f(g(x)) = 3\left(\frac{x}{3}\right) = x \] and \[ (g \circ f)(x) = g(f(x)) = \frac{3x}{3} = x. \]
Conclusion: Therefore, \( f \) and \( g \) are inverses of each other.
If \( f \) and \( g \) are inverses of each other, then the domain of \( f \) equals the range of \( g \), and the range of \( f \) equals the domain of \( g \).
Let \[ f(x) = \sqrt{x - 3}. \] The domain of \( f \) is \([3, +\infty)\), and its range is \([0, +\infty)\).
To find the inverse function, solve \( y = \sqrt{x - 3} \) by squaring both sides (\( y^2 = x - 3 \implies x = y^2 + 3 \)) and interchanging variables, giving: \[ f^{-1}(x) = x^2 + 3, \quad x \ge 0 \]
By Property 4, the domain of \( f^{-1} \) is \([0, +\infty)\), and its range is \([3, +\infty)\).
If \( f \) and \( g \) are inverses of each other, then their graphs are reflections of each other across the line \( y = x \).
Below are the graphs of \[ f(x) = \sqrt{x - 3} \] and its inverse \[ f^{-1}(x) = x^2 + 3, \quad x \ge 0. \]
If the point \( (a, b) \) lies on the graph of \( f \), then the point \[ (b, a) \] lies on the graph of the inverse function \( f^{-1} \).