ForHosting KIT · Math

Inverse interpolation calculator

Read a table backwards: with points (2, 4) and (3, 8) in your data, a target of 6 gives x of 2.5 by straight-line interpolation. Targets outside the table are refused, never guessed. Free in your browser.

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Run — free How it works & API

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Worked example

If you enter

Target y: 6

You get

X: 2.5
Solutions: 2.5
Solution count: 1
Y min: 2

Inverse interpolation answers the reverse of ordinary table lookup: you already know the dependent value y you want, and you need the independent abscissa x that would produce it. Calibration curves, growth tables, thermodynamics charts, and exam problems all publish (x, y) samples where the interesting unknown is x, not y. This inverse interpolation calculator accepts a list of finite sample pairs and a target y, sorts the table by y, brackets the target between neighbouring samples, and applies the classical linear formula that treats x as the dependent variable on that chord. Exact hits on a tabulated y return that node’s x without fraction noise. Targets outside the observed y range are rejected rather than extrapolated, so automation never invents values beyond the data.

What inverse interpolation means and how the linear estimate is formed

Ordinary interpolation estimates y at a known x by blending neighbouring tabulated y values. Inverse interpolation flips the roles: you fix a target y and estimate the x that would map to it under the same piecewise-linear model of the table. Geometrically, each consecutive pair of samples after sorting by y defines a straight chord in the (x, y) plane.

When the target coincides with a tabulated y, the answer is simply that sample’s x—no division is required. When the tabulated y values are not monotonic in the original order, sorting by y still produces a well-defined set of y-ordered chords; non-monotonic underlying functions can therefore yield more than one x for the same target, and this capability returns every distinct solution it finds, ordered ascending, with the primary field x set to the smallest.

Horizontal chords where two consecutive samples share the same y do not define a unique inverse for other targets; they only contribute when the target equals that shared y through the exact-match path. The method field always reports linear for this release, matching the classroom formula used in numerical analysis primers and engineering handbooks.

Where reading a table backwards helps

Laboratories invert calibration tables daily: a sensor reports a voltage or absorbance y and the analyst needs concentration or physical displacement x without re-fitting a full regression model for every one-off query. Process engineers read steam or refrigerant charts tabulated against temperature or pressure and need the abscissa that matches a measured ordinate. Students of numerical methods check homework on inverse linear interpolation against a deterministic oracle that rejects out-of-range targets instead of silently extrapolating.

Exam generators and LMS plugins can grade inverse-table items by comparing the learner’s x to the rounded solutions array, including multi-root cases when the table is non-monotonic. Keep tables modest, pin the rounded x in tests, and you get reproducible inverse estimates for notebooks, edge workers, and classroom tools without standing up a general computer-algebra stack.

What you can do with it

Sensor and calibration table inversion

Given discrete (reading, true value) or (x, sensor y) pairs, estimate the physical x that corresponds to a measured y without re-fitting a global model.

Homework and numerical-methods checks

Verify student inverse linear interpolation answers against a deterministic bracket-and-blend oracle that refuses out-of-range targets.

Process charts and tabulated property lookup

Invert engineering tables (temperature–property, load–deflection) to recover the abscissa matching a known ordinate in automation scripts.

FAQ

What is inverse interpolation?

It estimates the independent value x that yields a known dependent value y from tabulated (x, y) samples, by interpolating with x treated as the dependent variable on a y-bracket.

Which formula does this calculator use?

Linear inverse interpolation: x = x0 + (y − y0)·(x1 − x0)/(y1 − y0) on the y-sorted segment that brackets the target. Exact tabulated y hits return that node’s x.

Can there be more than one solution?

Yes. If the table is non-monotonic in y after sorting or multiple nodes share the target y, solutions lists every distinct x ascending; the field x is the smallest.

Which x matches a target of 6 between the points (2, 4) and (3, 8)?

It is 2.5. The target sits halfway between 4 and 8, so x sits halfway between 2 and 3. The calculator applies the linear formula on that segment of the sorted table.

What happens if my target y is outside the table?

The calculator stops with a clear message instead of extrapolating. The target must lie between the smallest and largest y in your data, so no value is ever invented beyond what you supplied.

Related tools

For developers — API access

Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.

API endpoint

POSThttps://api.kit.forhosting.com/calculus/inverse-interpolation

Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.

Call it from your stack

curl -X POST https://api.kit.forhosting.com/calculus/inverse-interpolation \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"points":[{"x":1,"y":2},{"x":2,"y":4},{"x":3,"y":8},{"x":4,"y":16}],"target_y":6}'

const res = await fetch("https://api.kit.forhosting.com/calculus/inverse-interpolation", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "points": [
      {
        "x": 1,
        "y": 2
      },
      {
        "x": 2,
        "y": 4
      },
      {
        "x": 3,
        "y": 8
      },
      {
        "x": 4,
        "y": 16
      }
    ],
    "target_y": 6
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/calculus/inverse-interpolation",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "points": [
        {
            "x": 1,
            "y": 2
        },
        {
            "x": 2,
            "y": 4
        },
        {
            "x": 3,
            "y": 8
        },
        {
            "x": 4,
            "y": 16
        }
    ],
    "target_y": 6
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/calculus/inverse-interpolation", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"points":[{"x":1,"y":2},{"x":2,"y":4},{"x":3,"y":8},{"x":4,"y":16}],"target_y":6}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"points":[{"x":1,"y":2},{"x":2,"y":4},{"x":3,"y":8},{"x":4,"y":16}],"target_y":6}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/calculus/inverse-interpolation", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "points": [
    {
      "x": 1,
      "y": 2
    },
    {
      "x": 2,
      "y": 4
    },
    {
      "x": 3,
      "y": 8
    },
    {
      "x": 4,
      "y": 16
    }
  ],
  "target_y": 6
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "calculus.inverse_interpolation",
  "status": "queued",
  "_links": {
    "result": "/tasks/tsk_…/result"
  }
}

The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.

Pricing

Per request$0.002

Published price — no tokens, no invented credits. A failed task is never charged.

Errors

HTTP	Code	Meaning
`401`	`unauthorized`	Missing or invalid API key.
`402`	`insufficient_balance`	Your balance doesn't cover the task price.
`404`	`unknown_type`	That task type doesn't exist.
`429`	`rate_limited`	Too many requests. Use the webhook instead of polling.

Technical notes

What inverse interpolation means and how the linear estimate is formed

If the target lies strictly between the two y-coordinates of that chord, the unique x on the line segment (or its extension along the same line between those endpoints’ y range) is given by x equals x0 plus (y minus y0) times (x1 minus x0) divided by (y1 minus y0).

Input contract, response shape, and validation rules

Pair the numeric result with your own plotting layer if you need a chart; this capability returns numbers and metadata only. POST a JSON body with points, a required array of at least two and at most five hundred objects each carrying finite numbers x and y (or [x, y] pairs, or a JSON string of the same list), and target_y, a finite number for the desired dependent value. Aliases data, samples, and table are accepted for points; aliases y, value, and target are accepted for target_y. Missing fields, empty lists, non-finite coordinates, and malformed objects fail with invalid_input naming the problem and are not billed. If target_y lies strictly outside the closed interval from the minimum to the maximum tabulated y, the task fails with a clear message that inverse interpolation requires the target inside the data range—this endpoint does not extrapolate beyond the table. On success the body echoes the cleaned target_y, the primary estimate x, the full solutions array and solution_count, method set to linear, exact_match true when a node was hit exactly, the primary segment with x0, y0, x1, and y1 used for display, y_min and y_max of the table, n_points, and a formula string restating the inverse linear identity. Example: points (1,2), (2,4), (3,8), (4,16) with target_y six brackets between y equals four and y equals eight, yielding x equals two point five because the chord from (2,4) to (3,8) rises by four while x advances by one, so half the y-rise from four to six is half a unit in x.

Technical questions

What if my target y is outside the table?

The task fails with invalid_input. This capability does not extrapolate beyond the minimum and maximum y in your points.

What does the inverse interpolation calculator cost?

The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.

Read the full KIT documentation →

Run — free

Worked example

What inverse interpolation means and how the linear estimate is formed

Where reading a table backwards helps

What you can do with it

Sensor and calibration table inversion

Homework and numerical-methods checks

Process charts and tabulated property lookup

FAQ

Related tools

For developers — API access

API endpoint

Call it from your stack

Example request

Example response

Pricing

Errors

Technical notes

What inverse interpolation means and how the linear estimate is formed

Input contract, response shape, and validation rules

Technical questions

thank you