Laura received her Master's degree in Pure Mathematics from Michigan State University, and her Bachelor's degree in Mathematics from Grand Valley State University. She has 20 years of experience teaching collegiate mathematics at various institutions.
Monotonic Function: Definition & Examples
Monotonicity
The monotonicity of a function tells us if the function is increasing or decreasing. A function is increasing when its graph rises from left to right. In technical terms, a function is increasing on an interval I if for any x1 and x2 in I, x1 is less than x2 implies that f(x1) is less than f(x2). Basically, this says that when x1 is less than x2, then the function evaluated at x1 is less than the function evaluated at x2.
A function is decreasing when its graph falls from left to right. Again, the technical definition says that a function is decreasing on an interval I if for any x1 and x2 in I, x1 is less than x2 implies that f(x1) is greater than f(x2). In other words, a function is decreasing on an interval I when it is the case that whenever x1 is less than x2, f(x1) is greater than f(x2).
The monotonicity of a function is directly related to the function's derivative. A function is increasing when its derivative is positive, and a function is decreasing when its derivative is negative. For example, consider the function f(x) = x 2. The derivative is f ' (x) equals 2x. Now consider where 2x is positive and negative.
2x is greater than 0.
Divide both sides by 2.
x is greater than 0.
Therefore, f is positive when x is greater than 0, so f is increasing on the interval x is greater than 0.
Similarly, consider where 2x is negative:
2x is less than 0
Divide both sides by 2
x is less than 0
Thus, f ' is negative when x is less than 0, and f is decreasing on the interval x is less than 0.
Montonic Function
When a function is increasing on its entire domain or decreasing on its entire domain, we say that the function is strictly monotonic, and we call it a monotonic function. For example, consider the function g(x) equals x 3:
Notice the graph of g is increasing everywhere. Therefore, this is a monotonic function. We can also observe this by looking at the derivative of g. The derivative of g(x) equals x 3 is g' (x) equals 3x 2. Notice that no matter what we plug in for x in the derivative, it gets squared, making it positive. Thus, the derivative is always positive, so g is always increasing. Therefore, g is a monotonic function.
Non-Monotonic Function
To solidify our understanding of what a monotonic function is, let's consider what a non-monotonic function is. A non-monotonic function is a function that is increasing and decreasing on different intervals of its domain.
For example, consider our initial example f(x) equals x 2. We saw that this function is increasing on the interval x is greater than 0, and decreasing on the interval x is less than 0. Since the function is increasing and decreasing on different intervals of its domain, the function is a non-monotonic function. Basically, if a function is not increasing on its entire domain or decreasing on its entire domain, then the function is not monotonic, and we say that it is non-monotonic.
Examples
1.) Consider the function h. Is this function monotonic?
Solution: Notice that the function is always going down from left to right. Thus, the function is decreasing on its entire domain. This tells us that the function is a monotonic function.
2.) Consider the function r(x) equals e x. Is this function monotonic?
Solution: This function is an interesting one, because its derivative is the same as the function. That is, the derivative of r(x) equals e x is r ' (x) equals e x. Notice, from the graph e x is always positive (it always lies above the y-axis). Therefore, the derivative of r is always positive. This tells us that r is always increasing (which can also be observed by its graph). Thus, r is a monotonic function.
Lesson Summary
An increasing function is a function that rises from left to right. Similarly, a decreasing function is a function that falls from left to right. A function's monotonicity refers to whether the function is increasing or decreasing. When a function is increasing on its entire domain or decreasing on its entire domain, we say that the function is monotonic.
We can determine if a function is monotonic by observing its graph, or we can check its derivative. A function is increasing when its derivative is positive, and it is decreasing when its derivative is negative.
Key Terms
Monotonicity - property of a function which determines if the function is increasing or decreasing
Increasing - when a function's graph rises from left to right
Decreasing - when a function's graph falls from left to right
Strictly monotonic - when a function is increasing on its entire domain or decreasing on its entire domain
Monotonic function - a function which graphs as strictly monotonic
Non-monotonic function - a function that is increasing and decreasing on different intervals of its domain
Learning Outcomes
The conclusion of the video lesson could correspond with your ability to:
- Define monotonicity
- Evaluate whether a function has increasing or decreasing monotonicity
- Review graphs to determine whether the represented function is monotonic or non-monotonic
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