Mathematics is full of complementary principles and symmetry. Perhaps nowhere is this more evident than with *inverse functions*: functions that “un-do” one another when put together. A few examples of inverse functions are shown in the following table:

$f(x)$ | $f^{-1}(x)$ |
---|---|

Addition | Subtraction |

Multiplication | Division |

Power | Root |

Exponential | Logarithm |

Derivative | Integral |

Inverse functions are vital to master if one hopes to be able to manipulate algebraic (literal) expressions. For example, to solve for time (\(t\)) in this exponential formula, you must know that the natural logarithm function directly “un-does” the exponential \(e^x\). This is the only way to “unravel” the equation and get \(t\) isolated by itself on one side of the equals sign:

\[V = 12e^{-t}\]

\[\hbox\textit{Divide both sides by 12}\]

\[{V \over 12} = e^{-t}\]

\[\hbox\textit{Take the natural logarithm of both sides}\]

\[\ln \left({V \over 12}\right) = \ln \left(e^{-t}\right)\]

\[\hbox\textit{The natural logarithm ``cancels out'' the exponential}\]

\[\ln \left({V \over 12}\right) = -t\]

\[\hbox\textit{Multiply both sides by negative one}\]

\[- \ln \left({V \over 12}\right) = t\]

In industry there exist a great many practical problems where inverse functions play a similar role. Just as inverse functions are useful for manipulating literal expressions in algebra, they are also useful in inferring measurements of things we cannot directly measure. Many continuous industrial measurements are *inferential* in nature, meaning that we actually measure some other variable in order to quantify the variable of interest. More often than not, the relationship between the primary variable and the inferred variable is nonlinear, necessitating some form of mathematical processing to complete the inferential measurement.

Practical examples of inferential measurement include:

- Liquid flow measurement through restrictions in pipes
- Liquid flow measurement in open channels (over weirs)
- Liquid level measurement in non-cylindrical vessels
- Temperature measurement by radiated energy
- Chemical composition measurement

The following sections will describe the mathematics behind each of these measurement applications.

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