Open the page and three panels appear. On the left is the function picker with eleven base functions. In the center is the plot panel with three curves: the function g(x) in blue, its inverse g−1(x) in amber, and a dashed gray line y=x that acts as the mirror across which g and g−1 reflect. On the right is the info panel with three tabs.
Below the picker sit four parameter sliders (a, k, b, h) that transform the base function. The page launches on the quadratic family — a classic example of a function that requires a restricted domain to be invertible.
Two header badges flag the current state. A yellow "domain restricted" badge appears for functions like quadratic, absolute value, sine, and cosine. A green "self-inverse (at defaults)" badge appears for functions like identity and reciprocal, which equal their own inverses when no transformation is applied.
Picking a Base Function
The picker lists eleven families, with sine and cosine grouped under "Trigonometric":
Functions marked with a small R badge in the picker are restricted — they fail the horizontal line test in their natural domain and need to be restricted to an invertible branch before the inverse can be defined. The visualizer shows both the full curve (faded) and the chosen branch (bold) for restricted families. The bottom of the picker spells out what R means.
Click any family to switch. Parameters reset to defaults on every switch, so you always start fresh.
Transforming the Function
The four sliders below the picker apply the standard affine transformations to the base function:
• a — vertical scale and reflection • k — vertical shift • b — horizontal scale and reflection • h — horizontal shift
The transformed function is g(x)=a⋅f(b(x−h))+k. The visualizer re-derives the inverse symbolically every time you move a slider, so g−1(x)=h+f−1((x−k)/a)/b updates in real time. Both equations are displayed as monospace badges in the plot header.
The most important thing this slider strip teaches: transforming f does not just move the inverse on the screen, it changes *which* transformations the inverse carries. A vertical scale of f becomes a horizontal scale of f−1. Vertical shifts on f become horizontal shifts on f−1. The Parameters tab in the info panel makes this explicit.
Reading the Plot
Up to four curves can appear in the plot at once:
• Gray dashed liney=x — the mirror line. Every point on g has a mirror point on g−1 across this line; their graphs are reflections of each other. • Blue solid curve — g(x). The transformed function. • Amber solid curve — g−1(x). The inverse, drawn only where it is defined. • Faded blue dashed curve — gfull (restricted families only). Shows the full base curve, with the bold blue curve highlighting the invertible branch.
For unrestricted families like cubic or exponential, only the three solid curves appear. For restricted families like quadratic, you see all four — the faded full parabola alongside the bold branch on x≥0, plus the inverse x in amber.
Crosshair and curve tooltips work the same as in other visualizers in the series — mouse over any curve to read off coordinates.
The Applied Chip Strip and Show Toggles
Two horizontal strips sit below the plot.
The Applied strip shows four chips, one per transformation parameter. Active (non-default) parameters glow blue with their current value. At a glance, you can tell which transformations are currently changing g — and therefore which mirrored transformations are affecting g−1.
The Show strip below has one toggle button per curve in the plot. Click a button to hide that curve. Hiding y=x removes visual clutter; hiding g−1 lets you focus on the function alone; hiding the full faded curve focuses you on just the invertible branch. The buttons preview the curve's color and line style (solid versus dashed) and show the curve's equation in monospace.
For self-inverse functions at default parameters, g and g−1 are the same curve and overlap exactly. Toggling either off shows that they were on top of each other.
The Info Panel — Three Tabs
The side info panel has three tabs:
• Explanation — reads the current state. Shows the base function and its inverse, the transformed equations, and special notes for the current family (self-inverse identity, restriction explanation for restricted families). Closes with the inverse-check identity g(g−1(x))=x.
• Parameters — explains the rule by which transformations of f become transformations of f−1. Includes a table showing each correspondence (vertical scale on f becomes horizontal scale on f−1, vertical shift becomes horizontal shift, and so on), then describes the current parameter values one by one. The most useful tab for understanding *why* the inverse changes the way it does.
• Concepts — general theory: reflection across y=x, the horizontal line test, restricted branches, the mirror identity. Independent of the current state.
What is an Inverse Function?
The inverse of a function f is the function f−1 that undoes f: if f(a)=b then f−1(b)=a. Every input-output pair (a,b) on the graph of f becomes the pair (b,a) on f−1 — the coordinates swap.
Geometrically, swapping coordinates is the reflection across the line $y = x$. The visualizer always draws this mirror line as a dashed gray reference, and you can verify that g and g−1 are mirror images of each other across it: pick any point on the blue curve, reflect it across y=x, and you will land on the amber curve.
The defining identity is the composition
g∘g−1(x)=xandg−1∘g(x)=x
on whichever domains both sides are defined.
The Horizontal Line Test and Restricted Branches
Not every function has a single-valued inverse. A function f is invertible only when no horizontal line crosses its graph more than once — the horizontal line test. If a horizontal line hits f twice, two different inputs produce the same output, and the inverse would have to map one input to two outputs, which is not allowed for a function.
Functions that fail the test can still be inverted on a restricted subdomain where they are strictly monotonic:
• Quadraticx2 restricted to [0,∞)→ inverse is x • Absolute value∣x∣ restricted to [0,∞)→ inverse is the identity on [0,∞) • Sine restricted to [−π/2,π/2]→ inverse is arcsin(x), defined on [−1,1] • Cosine restricted to [0,π]→ inverse is arccos(x), defined on [−1,1]
These are called principal branches. The visualizer shows the discarded portion as a faded curve and the kept portion as a bold curve, making the cut explicit and visible. When you transform a restricted function, the restriction boundary moves along with it.
How Transformations of f Become Transformations of f−1
Solving y=a⋅f(b(x−h))+k for x gives the inverse explicitly:
g−1(x)=h+b1⋅f−1(ax−k)
Reading the formula, each transformation on f has a mirrored counterpart on f−1 — when you swap axes (which is what reflecting across y=x does), vertical operations become horizontal and vice versa:
• Vertical scale a on f becomes horizontal scale 1/a on f−1 • Vertical shift k on f becomes horizontal shift k on f−1 • Horizontal scale b on f becomes vertical scale 1/b on f−1 • Horizontal shift h on f becomes vertical shift h on f−1
This is the same as the geometric fact that reflecting across y=x swaps horizontal and vertical directions. The visualizer's Parameters tab describes the active correspondences in plain English as you move the sliders, and the equation badges in the plot header show the consequence symbolically.
Functions That Are Their Own Inverse
Two picker families answer the inversion question with "already done": the identity and the reciprocal are self-inverse — applying them twice returns the input, so f−1=f.
Geometrically, self-inverse means the graph is its own mirror image across y=x. The identity lies *on* the mirror; the reciprocal straddles it symmetrically. In the plots for both, the amber inverse curve lands exactly on the blue one — the visualizer dashes the amber so the coincidence stays visible.
The Identity and Its Inverse
The identity f(x)=x is the fixed point of inversion itself: its graph is the mirror line, so reflecting changes nothing at all.
f(x) = x, frozen
Curve, inverse, and mirror in one line: the identity is the fixed point of reflection itself.
Blue curve, amber inverse, and gray mirror all coincide in one line — the degenerate case that makes the reflection rule legible everywhere else. Undoing "do nothing" is doing nothing.
Every self-inverse function has a graph symmetric across this line; the identity is simply the extreme case where the graph and the line are the same object.
The Scaled Linear Function and Its Inverse
Doubling is undone by halving: f(x)=2x has the inverse f−1(x)=x/2, the cleanest nontrivial inverse pair in the picker.
2x and x/2, frozen
Two lines with reciprocal slopes, reflecting into each other across the dashed mirror.
The two lines reflect across the mirror with slopes 2 and 21 — reciprocal slopes, which is the general rule for linear inverses: reflection across y=x swaps rise with run.
Every nonzero linear function is invertible with no restriction: one-to-one is automatic when a graph climbs (or falls) steadily. The lesson generalizes as monotonicity — the property the horizontal line test checks for.
The Cubic and the Cube Root
Cubing is fully undone by the cube root: both functions are defined on all of R, so this pair needs no restriction anywhere.
x³ and ∛x, frozen
A full inverse pair with no restriction anywhere: the cubic never turns around, so the cube root accepts every real input.
The cubic passes the horizontal line test despite its flat spot at the origin — it never actually turns around, so every output is hit exactly once. That is why 3x, unlike x, accepts negative inputs without complaint.
Compare the quadratic, one degree lower: the even power folds, fails the test, and pays with a restricted branch. Odd degree grants full invertibility; even degree costs half the domain.
The Reciprocal and Its Inverse
The reciprocal undoes itself: if y=1/x, then x=1/y — the same formula read backwards, making 1/x the picker's second self-inverse function.
1/x, self-inverse, frozen
The dashed amber curve retraces the blue hyperbola exactly: flipping a fraction twice restores it.
Both hyperbola branches are symmetric across the mirror line, each mapping onto itself under reflection. The dashed amber curve traces the blue one exactly.
Self-inversion here has an arithmetic reading: flipping a fraction twice restores it. The excluded point at x=0 survives inversion — the inverse (being the same function) excludes it too, in both domain and range.
The Exponential and the Logarithm
The exponential's inverse is the natural logarithm: ex and ln(x) reflect into each other across y=x with no restriction needed — the exponential is strictly increasing everywhere.
eˣ and ln(x), frozen
The reflection swaps the asymptotes: horizontal floor becomes vertical wall, domain and range trade axes.
The reflection swaps their asymptotes along with everything else: the exponential's horizontal floor at y=0 becomes the logarithm's vertical wall at x=0; domain R and range (0,∞) trade places exactly.
This is the pair to study for the domain-range swap rule: whatever holds for f on one axis holds for f−1 on the other. The logarithmic entry shows the same picture from the other side.
The Logarithm and the Exponential
Choosing the logarithm as the base function shows the same inverse pair from the other direction: now ln(x) is blue and ex is amber.
ln(x) and eˣ, frozen
The same pair viewed from the other end — now the blue curve owns the wall and the amber one the floor.
Nothing mathematical changes — inversion is symmetric, (f−1)−1=f — but the visual swap is worth one deliberate look: the blue curve now owns the vertical asymptote and the restricted domain, the amber one the horizontal floor.
That interchangeability is the point: an inverse pair is one relationship viewed from two ends, not two separate facts. The exponential entry is this section's mirror twin, in every sense.
The Quadratic and the Square Root
The parabola fails the horizontal line test — every positive output comes from two inputs — so inverting it requires surgery: keep only the branch x≥0, shown bold over the faded full curve.
x² restricted, and √x, frozen
The faded left arm is the price of invertibility: only the bold right branch reflects into the square root.
The kept right half is one-to-one, and its reflection is exactly x — the amber curve. The discarded left half is why x returns only the non-negative root: the branch choice made here *is* the definition of the square root function.
Choosing the other branch (x≤0) would have produced −x instead. Branch choice is a genuine convention, standardized so that everyone's square root agrees.
This is the template for every restricted family: fold detected by the horizontal line test, branch kept, reflection taken.
The Square Root and Its Inverse
Starting from x and inverting lands back on the parabola — but only its right half: f−1(x)=x2 restricted to x≥0.
√x and its inverse, frozen
Inverting the root lands on the parabola’s right half only — the restriction is inherited from the root’s range.
The restriction is inherited, not imposed: the square root's range is [0,∞), so its inverse's domain can be no larger. Inverting never recovers what a restriction already discarded — the left half of the parabola is gone for good.
Together with the quadratic entry, this pair shows both directions of the same relationship, and why (x)2=x holds on x≥0 while x2=∣x∣ does not reduce to x — the asymmetry the composition tool demonstrates as its star example.
The Absolute Value and Its Inverse
The V fails the horizontal line test as badly as a function can — every positive output has exactly two preimages — so only the branch x≥0 survives, where ∣x∣ is simply x.
|x| restricted, frozen
The kept branch of the V is just y = x, so branch, inverse, and mirror coincide; the faded arm is the discarded half.
The kept branch coincides with the identity on the half-line, so its inverse does too: blue branch, amber inverse, and mirror line all overlap for x≥0. The faded left arm is the discarded half.
The absolute value makes the restriction lesson stark: what remains after cutting to a branch can be almost trivial. The interesting content was in the fold itself — and the fold is exactly what one-to-one functions are forbidden to have.
Sine and the Arcsine
A periodic wave hits every output infinitely often — the worst possible horizontal line test failure — so sine is restricted to one increasing sweep, [−π/2,π/2], before inverting.
sin(x) on [−π/2, π/2], and arcsin, frozen
One bold rising arch of the faded wave reflects into arcsine: domain [−1, 1], principal values on the y-axis.
That single bold arch, rising from −1 to 1, reflects into arcsin(x): a curve living on the domain [−1,1] (sine's range, swapped into position) and producing angles in [−π/2,π/2] — the *principal values*.
Every other period of the faded wave could have been chosen; the standardized branch is the one crossing the origin. Calculators answer arcsin(0.5)=30° rather than 150° because of exactly this convention.
Cosine's branch solves the same problem with a different cut — comparing the two shows how branch choice adapts to each function's shape.
Cosine and the Arccosine
Cosine's restriction cannot copy sine's: on [−π/2,π/2] cosine rises and falls, failing the test inside the window. The invertible sweep is [0,π] instead — one full descent from 1 to −1.
cos(x) on [0, π], and arccos, frozen
Cosine’s cut is one falling sweep, so arccosine falls too — from π down to 0 across its [−1, 1] domain.
Reflecting that falling branch produces arccos(x): defined on [−1,1], returning angles in [0,π], and *decreasing* — the visible signature that distinguishes it from arcsine at a glance.
The two arc functions are tied by arccos(x)=π/2−arcsin(x): complementary branches of complementary functions.
Together with sine, this closes the picker's tour of restriction strategies: same disease (periodicity), same cure (one monotone sweep), different incision points.
Related Concepts and Tools
Functions Families Gallery — companion gallery showing the same base functions used here, useful as a prerequisite for understanding what each family looks like before studying its inverse.
Function Transformations — visualizer for the four affine transformations alone, without the inverse overlay. Helpful for building intuition before adding the inverse layer here.
Composition of Functions — the operation behind the inverse identity g∘g−1=id. Inverses are defined precisely as compositional partners.
One-to-One Functions — the formal property a function must have to be invertible. Equivalent to passing the horizontal line test.
Inverse Trigonometric Functions — focused treatment of arcsin, arccos, arctan, and the standard principal-branch conventions used here.
Logarithm and Exponential — paired study of the canonical example of inverse functions, with ln and ex as exact inverses.
Derivative of an Inverse Function — the rule (f−1)′(x)=1/f′(f−1(x)); a calculus follow-up to the geometric reflection studied here.