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Inverse functions

An inverse function reverses a one-to-one function by assigning each output its unique original input.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. If f(x)=2x+5, then input 3 produces output 11
  3. The superscript -1 names the inverse rule; it does not mean the reciprocal 1/f(x)
  4. Check your understanding
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01If f(x)=2x+5, then input 3 produces output 11#

If f(x)=2x+5f(x)=2x+5, then input 3 produces output 11. To recover that input from 11, undo the addition first, then the multiplication.

To invert f(x)=2x+5 at output 11, move horizontally from height 11 to the curve and then down to input 3. Algebraically (11−5)/2=3.To invert f(x)=2x+5 at output 11, move horizontally from height 11 to the curve and then down to input 3. Algebraically (11−5)/2=3.
Figure 1To invert f(x)=2x+5 at output 11, move horizontally from height 11 to the curve and then down to input 3. Algebraically (11−5)/2=3.
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For f(x)=2x+5f(x)=2x+5, recover the input for output 11: 11−5=611-5=6, then 6/2=36/2=3. We write f−1(11)=3f^{-1}(11)=3, read “f inverse of 11.” The forward check is f(3)=11f(3)=11.

Check your reasoning

For f(x)=2x+5f(x)=2x+5, what is f−1(9)f^{-1}(9)?

Show answer and explanation
2

Recover the input: (9−5)/2=2(9-5)/2=2. The forward check is f(2)=9f(2)=9.

02The superscript -1 names the inverse rule; it does not mean the reciprocal 1/f(x)#

The superscript −1-1 names the inverse rule; it does not mean the reciprocal 1/f(x)1/f(x). An inverse needs a unique original input. A rule that sends two different inputs to the same output cannot reverse that output uniquely.

Check your reasoning

A one-to-one function has g(4)=9g(4)=9. Which statement follows?

  1. Ag−1(4)=9g^{-1}(4)=9
  2. Bg−1(9)=4g^{-1}(9)=4
  3. Cg−1(9)=1/9g^{-1}(9)=1/9
Show answer and explanation
g−1(9)=4g^{-1}(9)=4

The inverse maps output 9 back to input 4.

03Check your understanding#

Check your reasoning

A reversible calibration multiplies raw input by 3, then subtracts 1. A calibrated output is 14. What was the raw input?

Show answer and explanation
5

Undo subtraction: 14+1=1514+1=15. Undo multiplication: 15/3=515/3=5.

Key takeaway

Recover the input by undoing the operations in reverse order.

  • Recover an input using an invertible scalar function

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Inverse functions.” Math behind ML. /learn/f-inverse