Invertible Functions

An invertible function is a function that can be 'undone' — given any output, you can uniquely recover the input that produced it. Just as multiplication by 3 is undone by division by 3, a function f is undone by its inverse function, written f⁻¹. The inverse takes every output of f back to the original input, so f⁻¹(f(x)) = x and f(f⁻¹(y)) = y for all valid inputs. Not every function has an inverse. If two different inputs produce the same output — for example, f(x) = x² sends both 3 and −3 to 9 — there is no way to decide which input to return to, so the inverse cannot be a function. A function must be one-to-one (injective) to be invertible: each output comes from exactly one input. Graphically, a function is one-to-one when every horizontal line crosses its graph at most once — the Horizontal Line Test. When a function fails the Horizontal Line Test over its natural domain, mathematicians restrict the domain to a smaller interval on which the function is one-to-one, then define the inverse only on that restricted version. This technique is essential in practice: for instance, the square-root function √x is defined as the inverse of x² restricted to x ≥ 0, and the trigonometric inverse functions sin⁻¹, cos⁻¹, and tan⁻¹ all arise from carefully chosen domain restrictions.

Key takeaways
  • Check that the function is one-to-one
  • Swap x and y, then solve for y
  • State the domain and range of the inverse
  • Verify using composition
  • Interpret the graph

Check that the function is one-to-one

Apply the Horizontal Line Test to the graph, or verify algebraically that f(a) = f(b) implies a = b. If the function is not one-to-one on its natural domain, identify a restricted domain (e.g., x ≥ 0) on which it is. Every strictly increasing or strictly decreasing function is automatically one-to-one.

Swap x and y, then solve for y

Write the function as y = f(x). Exchange every x with y and every y with x to get x = f(y). Solve the resulting equation for y. The expression you obtain is f⁻¹(x). For example, starting with y = 2x + 5, swapping gives x = 2y + 5, so y = (x − 5)/2, meaning f⁻¹(x) = (x − 5)/2.

State the domain and range of the inverse

The domain of f⁻¹ equals the range of f, and the range of f⁻¹ equals the domain of f. If you restricted the domain of f (say, to x ≥ 0), then f⁻¹ has that restricted set as its range. Always write both the formula and the domain when presenting an inverse function.

Verify using composition

Confirm the inverse is correct by checking both composite identities: f⁻¹(f(x)) = x for every x in the domain of f, and f(f⁻¹(x)) = x for every x in the domain of f⁻¹. If either check fails, the inverse formula or domain is wrong. This double-check is not optional — it catches algebra errors and domain mistakes.

Interpret the graph

The graph of f⁻¹ is the reflection of the graph of f across the line y = x. Every point (a, b) on the graph of f corresponds to the point (b, a) on the graph of f⁻¹. This geometric symmetry gives a powerful visual check: if you fold the coordinate plane along the line y = x, the two graphs should coincide exactly.

The Key Equation

f−1(f(x))=xandf(f−1(y))=yf^{-1}\bigl(f(x)\bigr) = x \quad \text{and} \quad f\bigl(f^{-1}(y)\bigr) = y
  • f — Original function (none): A one-to-one function with domain D and range R.
  • f^{-1} — Inverse function (none): The function with domain R and range D that reverses the action of f. Note: f⁻¹(x) is NOT 1/f(x).
  • x — Input of f (none): Any element in the domain D of f.
  • y — Input of f⁻¹ (none): Any element in the domain of f⁻¹, which equals the range R of f.

Worked Example

Problem. Let f(x) = x² − 4 with domain x ≥ 0. Find f⁻¹(x), state its domain and range, and verify using composition.

f−1(x)=x+4,x≥−4f^{-1}(x) = \sqrt{x+4}, \quad x \geq -4

Answer. f⁻¹(x) = √(x + 4), domain x ≥ −4, range y ≥ 0.

Figure
Diagram
xyy = xf(x)=x²−4, x≥0f⁻¹(x)=√(x+4)(a,b)↔(b,a)−4−40Graph of f and f⁻¹ reflected across y = x
Origins · 1 minLeonhard Euler and how this was discovered
LE
Leonhard Euler
1707–1783 · Swiss

Euler introduced the modern notation f(x) for functions in 1734 and systematically developed the concept of a function as a correspondence between quantities. His work laid the groundwork for rigorously defining inverses: because he clarified what a function is, later mathematicians could precisely state when and how it can be reversed. Euler also worked extensively with inverse trigonometric and logarithmic functions. Euler lost sight in his right eye in 1738 and became almost completely blind by 1771, yet he continued to produce mathematics at an astonishing rate — nearly half of his total published output came after he lost most of his vision, dictated to assistants from memory.

  • One-to-One (Injective) Functions — A function must be one-to-one to have an inverse; understanding injectivity is the essential prerequisite for invertibility.
  • Composition of Functions — The defining property of an inverse — f⁻¹(f(x)) = x — is stated using composition, so you must understand function composition to verify inverses.
  • Domain and Range — The domain and range of a function swap when you form its inverse; domain restrictions are the key tool for making non-invertible functions invertible.
  • Inverse Trigonometric Functions — sin⁻¹, cos⁻¹, and tan⁻¹ are the most important real-world examples of inverses defined through domain restriction.
  • Logarithms — The logarithm log_b(x) is defined as the inverse of the exponential b^x, making it a major application of invertible functions.