Which of the following are monotonic transformations? (1)...

Which of the following are monotonic transformations?
(1) $u=2 v-13$
(2) $u=-1 / v^{2}$
(3) $u=1 / v^{2}$
(4) $u=\ln v$
(5) $u=-e^{-v}$
$(6) \quad u=v^{2}$
(7) $u=v^{2}$ for $v>0 ;$ (8) $u=v^{2}$ for $v<0$.

Key Concepts

Monotonic Transformation

A monotonic transformation is a function that preserves the order of the data points it is applied to. In other words, if one value is larger than another before the transformation, it remains so after the transformation. This is a crucial concept in statistics and mathematics because it ensures the relative ranking of values is maintained, which is important in many applications such as non-parametric statistics and ordinal data analysis.

Strictly Increasing Function

A strictly increasing function is one in which every increase in the input value results in an increase in the output value. Formally, for any two inputs, if one is greater than the other, then the output of the first is also greater than the output of the second. This property guarantees the preservation of order and is an essential characteristic of many monotonic transformations.

Strictly Decreasing Function

A strictly decreasing function is one in which every increase in the input value results in a decrease in the output value. In this case, if one input is greater than the other, then the corresponding output will be smaller. Although the order is reversed relative to the input, the function is still monotonic because the relative ordering is consistently maintained. This concept is important when analyzing functions that might invert relationships while preserving monotonicity.

Domain Restrictions

Domain restrictions refer to the specific set of input values for which a function is defined. In the context of monotonic transformations, the domain is critical because a function might be monotonic over a restricted interval even if it is not monotonic over its entire natural domain. For example, certain quadratic functions are only monotonic when limited to a half-line. Understanding the role of domain restrictions is essential for correctly applying and interpreting monotonic transformations.

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