Question

The text said that raising a number to an odd power was a monotonic transformation. What about raising a number to an even power? Is this a monotonic transformation? (Hint: consider the case \(f(u)=u^{2}\).)

Short answer

Raising a number to an even power is not a monotonic transformation.

Step-by-step solution

Understanding Monotonic Transformations

A function is said to be monotonic if it is either entirely non-increasing or non-decreasing. This means that as you increase the input, the output should also increase or stay the same (non-decreasing) or decrease or stay the same (non-increasing).

Exploring Odd Powers

When a number is raised to an odd power, the function is monotonic. For example, consider the function \( g(u) = u^3 \). As \( u \) increases or decreases, \( g(u) \) consistently increases or decreases respectively. Thus, odd power functions retain the same directionality.

Considering Even Powers

Now, let's explore the function \( f(u) = u^2 \). If \( u \) is positive, then increasing \( u \) results in increasing \( u^2 \). However, if \( u \) is negative, increasing \( u \) (i.e., moving from a more negative value to a less negative value) results in decreasing \( u^2 \). Therefore, \( f(u) \) does not consistently increase or decrease.

Analyzing the Behavior at Zero

At \( u = 0 \), the function \( f(u) = u^2 \) achieves its minimum value (0), since squares of all numbers result in a non-negative number. As \( u \) moves away from zero in either direction (positive or negative), \( f(u) \) increases.

Conclusion on Monotonicity

Since \( f(u) = u^2 \) does not exhibit consistent increase or decrease over the entire domain (negative and positive values of \( u \)), it is not a monotonic transformation. The behavior changes as \( u \) crosses zero, which is not characteristic of a monotonic function.

Key concepts

Odd powers

Odd powers apply to expressions where numbers are raised to an odd exponent, such as 1, 3, 5, etc. Functions using odd powers, like \( g(u) = u^3 \), exhibit monotonic behavior. This means that

• if the input \( u \) increases, the function \( g(u) \) also increases.
• if the input \( u \) decreases, \( g(u) \) decreases along with it.

This type of function retains the directionality of the input. In other words, it maintains a consistent path of either increasing or decreasing without switching back. Think of it as a steady climb up or descent down a hill, where there are no plateaus or turnarounds. Thanks to this property, odd powers are considered monotonic transformations because they don't alter the basic 'direction' of a function. This makes them predictable and easier to analyze in terms of changes in input value.
Functions involving odd powers are particularly useful in math for describing relationships where consistent directional change is important. It helps in visualizing how values behave across both positive and negative domains.

Even powers

Even powers refer to expressions raised to an even exponent, such as 2, 4, 6, etc. A typical example is the quadratic function \( f(u) = u^2 \). These functions do not exhibit a monotonic pattern. Here’s how they behave:

• For positive \( u \), increasing \( u \) results in increasing \( u^2 \).
• For negative \( u \), increasing \( u \) results in decreasing \( u^2 \).

Therefore, there is a shift in behavior depending on whether \( u \) is positive or negative. At the moment when \( u \) crosses zero, the function achieves its lowest point, giving us a U-shaped curve. This total structure means the function is not consistently increasing or decreasing over its entire domain.
Consider it like a valley, where the lowest point is at zero. As the value of \( u \) moves away from zero, whether in a positive or negative direction, \( f(u) \) increases in both cases, unlike odd powers. Thus, we cannot call these transformations monotonic. This behavior is particularly important in differential calculus and analysis of function behaviors, especially in optimization and stability.

Non-decreasing function

A non-decreasing function is one where the output either increases or stays the same as the input increases. It never decreases. Importantly, this does not mean that it has to increase, it just must not decrease. This concept is essential for understanding monotonicity.

• In mathematical terms, a function \( f \) is non-decreasing if, for every \( x_1 \leq x_2 \), \( f(x_1) \leq f(x_2) \).

For example, the function \( h(u) = u^3 \) is both non-decreasing and non-increasing, depending on which direction you look at from a starting point. It reflects the property of odd powers. On the other hand, even powers show us that a function is not non-decreasing if it has areas where outputs decrease as inputs increase, like with negative values in \( f(u) = u^2 \).
Understanding non-decreasing functions helps in predicting and explaining consistent output behavior in mathematical contexts. This knowledge is valuable when interpreting graphs and analyzing data sets to ensure they maintain a certain trend or direction over a specified range. Recognizing whether a function is non-decreasing aids in many trades like economics, physics, and data science. It makes it clear when a function preserves order, which is crucial for mapping out steady growth or decay trends in practical applications.