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An inverse function f-1 reverses the action of f: if f(a) = b, then f-1(b) = a. A function has an inverse if and only if it is one-to-one (passes the horizontal line test). The domain of f becomes the range of f-1 and vice versa. Graphically, f and f-1 are reflections across the line y = x. Inverse functions are the foundation for logarithms, inverse trig functions, and solving equations.
Start with simple examples (linear functions), find inverses algebraically by swapping x and y and solving, then verify by composing f(f-1(x)) = x. Use the horizontal line test to determine invertibility, and discuss restricting domains to create invertible functions.
You know from your study of function notation that a function f takes an input x and produces an output f(x). An inverse function f⁻¹ does exactly the reverse: it takes the output of f and recovers the original input. Formally, if f(a) = b, then f⁻¹(b) = a. This is not about reciprocals — f⁻¹(x) is not 1/f(x) — it's about undoing. If f multiplies by 3, then f⁻¹ divides by 3. If f adds 5, then f⁻¹ subtracts 5. If f takes a square (with appropriate domain), f⁻¹ takes a square root.
The critical question is: when does an inverse exist? Here, your knowledge of domain and range becomes essential. A function can only be inverted if it is one-to-one: every output comes from exactly one input. If two different inputs produced the same output, which one would the inverse recover? It cannot do both. The geometric test is the horizontal line test — if any horizontal line crosses the graph more than once, the function is not one-to-one and has no inverse (over that full domain). The standard function f(x) = x² fails this test over all reals (since f(2) = f(−2) = 4), but succeeds if you restrict the domain to x ≥ 0 — which is exactly how the square root function is defined as the inverse of the "right half" of the parabola.
To find the inverse algebraically: write y = f(x), then swap x and y (since the inverse reverses the roles of input and output), and solve for y. That expression in y is f⁻¹(x). The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f — they swap. You can always verify: the composition f(f⁻¹(x)) = x and f⁻¹(f(x)) = x should both hold (wherever defined). This composition identity is the *definition* of what it means for two functions to be inverses of each other.
Graphically, f and f⁻¹ are reflections of each other across the line y = x. This is because swapping x and y in the equation is precisely the algebraic description of reflecting across y = x. If the point (2, 5) is on the graph of f, then (5, 2) is on the graph of f⁻¹. This visual symmetry is a powerful check: if the graph of f doesn't look like a reflection of the graph of f⁻¹ across y = x, something has gone wrong.
This topic is the conceptual prerequisite for two major upcoming ideas. Logarithms are the inverse functions of exponentials — log_b(x) asks "what power of b gives x?" — and inverse trigonometric functions like arcsin, arccos, arctan are inverses of the trig functions restricted to appropriate domains. Every time you solve an equation by "undoing" an operation — taking a log of both sides, applying arcsin to isolate an angle — you are using inverse functions. Understanding the one-to-one requirement now prevents confusion later about why arcsin(sin(x)) ≠ x for all x.