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:
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 line $y = x$ — 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 — $g$ full (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:
• Quadratic $x^2$ 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.
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.