riaology
Unit 1 · Lesson 2 of 8

Undo a measurement with an inverse.

A sensor reports a squared displacement. To recover displacement, you must know which sign of the original input is allowed.

Change the input. Follow the mathematics.

What you'll be able to do
  • Set up the model and calculate its recovered nonnegative input.
  • Connect the equation to the changing visual.
  • Explain the assumptions and identify a misleading interpretation.
Predict before exploring

What will change if you increase measured output? Predict the direction, then explain the response.

A model you can changeMove one control at a time.
Read the modelf(x)=x2 (x≥0),f−1(y)=yf(x)=x^2\ (x\ge0),\quad f^{-1}(y)=\sqrt y

Restrict the square function to nonnegative inputs so it is one-to-one.

Input x versus Output x². Square, Inverse, Reflection line y=x. Selected recovered nonnegative input: 3 . The inverse recovers the input. Without the domain restriction, positive outputs would have two candidate inputs.0012.512.52525Output x²Input x
SquareInverse · dashedReflection line y=x · dashed

recovered nonnegative inputf−1(9)=9=3f^{-1}(9)=\sqrt{9}=3

The inverse recovers the input. Without the domain restriction, positive outputs would have two candidate inputs.

Assumptions, units, and limits

Restrict the square function to nonnegative inputs so it is one-to-one.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

01

From the situation to an equation

A function has an inverse on a domain where it is one-to-one. Swap the roles of input and output, solve for the old input, and carry the domain restriction with you. Function composition can verify the result.

f(x)=x2 (x≥0),f−1(y)=yf(x)=x^2\ (x\ge0),\quad f^{-1}(y)=\sqrt y
02

Keep the reasoning visible

f−1(f(x))=x(x≥0)f^{-1}(f(x))=x\quad(x\ge0)

The inverse recovers the input. Without the domain restriction, positive outputs would have two candidate inputs.

03

What makes this model useful?

Restrict the square function to nonnegative inputs so it is one-to-one.

An inverse function is not the reciprocal 1/f. Without restricting the square function, one reading can have two possible inputs.

A worked example

Calculate the starting scenario

  1. Use measured output = 9 m.
  2. f−1(9)=9=3f^{-1}(9)=\sqrt{9}=3
  3. The inverse recovers the input. Without the domain restriction, positive outputs would have two candidate inputs. The checked quantity is recovered nonnegative input.
Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the recovered nonnegative input using measured output 9 m. State the meaning of its sign and units.

Checked quantity: recovered nonnegative input · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

A function has an inverse on a domain where it is one-to-one. Swap the roles of input and output, solve for the old input, and carry the domain restriction with you. Function composition can verify the result.

Explore the visual solution
Read the modelf(x)=x2 (x≥0),f−1(y)=yf(x)=x^2\ (x\ge0),\quad f^{-1}(y)=\sqrt y

Restrict the square function to nonnegative inputs so it is one-to-one.

Input x versus Output x². Square, Inverse, Reflection line y=x. Selected recovered nonnegative input: 3 . The inverse recovers the input. Without the domain restriction, positive outputs would have two candidate inputs.0012.512.52525Output x²Input x
SquareInverse · dashedReflection line y=x · dashed

recovered nonnegative inputf−1(9)=9=3f^{-1}(9)=\sqrt{9}=3

The inverse recovers the input. Without the domain restriction, positive outputs would have two candidate inputs.

Assumptions, units, and limits

Restrict the square function to nonnegative inputs so it is one-to-one.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

A function has an inverse on a domain where it is one-to-one. Swap the roles of input and output, solve for the old input, and carry the domain restriction with you. Function composition can verify the result.

f(x)=x2 (x≥0),f−1(y)=yf(x)=x^2\ (x\ge0),\quad f^{-1}(y)=\sqrt y
Calculate and explain

Apply the idea to a new scenario

A sensor reading is 16 m on this nonnegative-input model. Recover the input.

Checked quantity: recovered nonnegative input · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

A function has an inverse on a domain where it is one-to-one. Swap the roles of input and output, solve for the old input, and carry the domain restriction with you. Function composition can verify the result.

Explore the visual solution
Read the modelf(x)=x2 (x≥0),f−1(y)=yf(x)=x^2\ (x\ge0),\quad f^{-1}(y)=\sqrt y

Restrict the square function to nonnegative inputs so it is one-to-one.

Input x versus Output x². Square, Inverse, Reflection line y=x. Selected recovered nonnegative input: 4 . The inverse recovers the input. Without the domain restriction, positive outputs would have two candidate inputs.0012.512.52525Output x²Input x
SquareInverse · dashedReflection line y=x · dashed

recovered nonnegative inputf−1(16)=16=4f^{-1}(16)=\sqrt{16}=4

The inverse recovers the input. Without the domain restriction, positive outputs would have two candidate inputs.

Assumptions, units, and limits

Restrict the square function to nonnegative inputs so it is one-to-one.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

A function has an inverse on a domain where it is one-to-one. Swap the roles of input and output, solve for the old input, and carry the domain restriction with you. Function composition can verify the result.

f(x)=x2 (x≥0),f−1(y)=yf(x)=x^2\ (x\ge0),\quad f^{-1}(y)=\sqrt y
Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual
Read the modelf(x)=x2 (x≥0),f−1(y)=yf(x)=x^2\ (x\ge0),\quad f^{-1}(y)=\sqrt y

Restrict the square function to nonnegative inputs so it is one-to-one.

Input x versus Output x². Square, Inverse, Reflection line y=x. Selected recovered nonnegative input: 3 . The inverse recovers the input. Without the domain restriction, positive outputs would have two candidate inputs.0012.512.52525Output x²Input x
SquareInverse · dashedReflection line y=x · dashed

recovered nonnegative inputf−1(9)=9=3f^{-1}(9)=\sqrt{9}=3

The inverse recovers the input. Without the domain restriction, positive outputs would have two candidate inputs.

Assumptions, units, and limits

Restrict the square function to nonnegative inputs so it is one-to-one.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. A function has an inverse on a domain where it is one-to-one. Swap the roles of input and output, solve for the old input, and carry the domain restriction with you. Function composition can verify the result.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

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