Functions & Their Graphs

Inverse Functions

Finding and verifying the inverse of a function algebraically and graphically.

Advanced20 min lesson4 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Composing Functions

The Function That Undoes Another

Recall from Algebra II that a logarithm undoes exponentiation — feed a value through an exponential function, then through the matching logarithm, and you get back exactly where you started. Before reading on, think about how you'd test, in general, whether two functions truly undo each other, for any pair of functions at all, not just logs and exponentials.

Definition — Inverse Function

f⁻¹ is the inverse of f if composing them in either order returns the original input: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for every x in the appropriate domain. Graphically, f⁻¹'s graph is f's graph reflected across the line y = x.

That reflection isn't a separate fact — it comes directly from how an inverse is found algebraically. Finding f⁻¹ means swapping x and y in f's equation and solving for the new y, and swapping x and y in an equation is exactly what reflecting a graph across y = x does geometrically. The algebra and the picture describe the same move.

Not every function has an inverse, though. A function assigns exactly one output to each input — but an inverse needs to reliably send each output back to exactly one input. If two different inputs shared the same output under f, the inverse wouldn't know which one to return to. A function only has a genuine inverse if it's one-to-one: every output comes from exactly one input, checked visually with the horizontal line test (no horizontal line crosses the graph more than once).

Worked Example — Finding and Verifying an Inverse

Find the inverse of f(x) = 2x + 3. Write y = 2x + 3, swap x and y: x = 2y + 3, then solve for y: y = (x − 3)/2. So f⁻¹(x) = (x − 3)/2. Verify: f(f⁻¹(x)) = 2 · [(x − 3)/2] + 3 = (x − 3) + 3 = x. And f⁻¹(f(x)) = [(2x + 3) − 3]/2 = 2x/2 = x. Both directions return x, confirming the inverse.

Worked Example — Testing for a Genuine Inverse with the Horizontal Line Test

Does f(x) = x² have an inverse over all real numbers? A horizontal line like y = 4 crosses the graph twice, at x = 2 and x = −2 — both inputs share the same output, so f isn't one-to-one and has no genuine inverse over its full domain. By contrast, f(x) = x³ passes the horizontal line test everywhere (every horizontal line crosses exactly once), so it does have a genuine inverse: f⁻¹(x) = ∛x.

Function Explorer

f(x) =

sin(x)

Transform: g(x) = a·f(b(x − h)) + k

a — vertical stretch/reflect1

b — horizontal stretch/reflect1

h — horizontal shift0

k — vertical shift0

Composition

Compose with another function

Trace mode

Show inverse view

Analysis (of the transformed function, in view)

y-intercept: (0, 0)
x-intercepts: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema: local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-1, 1]

Tip

The swap-x-and-y method for finding an inverse works because it's literally solving for the reflected relationship — after solving, double-check by composing the result with the original function in both directions, confirming it truly returns x.

Common Mistakes

Writing f⁻¹(x) as 1/f(x), confusing the inverse function notation with a reciprocal.
f⁻¹(x) means the inverse function, found by undoing f's operations — it's unrelated to 1/f(x), which is a completely different expression (the reciprocal of f's output).
Assuming every function has an inverse, without checking the horizontal line test (or restricting the domain) first.
A function needs to be one-to-one to have a genuine inverse — check the horizontal line test, or note whether the domain has been restricted to make it one-to-one, before finding an inverse.

Key Takeaways

An inverse function undoes the original: composing f and f⁻¹ in either order returns the original input.
Reflecting a graph across y = x corresponds exactly to swapping x and y in its equation, the algebraic method for finding an inverse.
A function needs to be one-to-one (passing the horizontal line test) to have a genuine inverse.

Summary

Inverses reverse a function's action, both algebraically and as a reflection across y = x. The next lesson studies a broader family of graph-changing moves — transformations — that shift, stretch, and reflect a function's graph in controlled ways.

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