Decreasing Function

Definition

A decreasing function is a type of function where, as the input values increase, the output values decrease. This behavior can be observed on a graph where the line moves downward from left to right. Decreasing functions are essential for understanding the overall shape and behavior of graphs, particularly in identifying intervals where a function is not increasing and analyzing the overall trend of the function.

5 Must Know Facts For Your Next Test

1. A function is considered decreasing on an interval if for any two points within that interval, if x1 < x2 then f(x1) > f(x2).
2. Decreasing functions can be identified using the first derivative test; if the derivative is negative over an interval, the function is decreasing there.
3. In a graph of a decreasing function, any horizontal line drawn will intersect the curve at most once, indicating that it does not repeat any output values.
4. Common examples of decreasing functions include linear functions with a negative slope and certain polynomial functions.
5. Decreasing functions can be classified as strictly decreasing (where the function decreases continuously) or non-strictly decreasing (where it may have flat sections).

Review Questions

• How can you determine whether a function is decreasing on a given interval using its derivative?
  • To determine if a function is decreasing on a specific interval, you can analyze its first derivative. If the derivative of the function is negative for all x-values in that interval, it indicates that the function itself is decreasing there. This means that as you move from left to right along the x-axis, the corresponding y-values will consistently drop.
• What is the significance of critical points in relation to identifying decreasing functions on a graph?
  • Critical points are significant because they represent where the function's behavior may change. At these points, the derivative may be zero or undefined, suggesting potential local maxima or minima. By examining these critical points alongside intervals defined by them, you can determine where the function is decreasing or increasing by observing changes in the sign of the derivative.
• Compare and contrast strictly decreasing functions with non-strictly decreasing functions and their implications in calculus.
  • Strictly decreasing functions consistently drop without flat sections, meaning for any two distinct x-values in their domain, the corresponding y-values will always be different and follow f(x1) > f(x2). Non-strictly decreasing functions may have flat segments where output values can repeat. This distinction affects how we analyze functions in calculus; understanding whether a function is strictly or non-strictly decreasing helps predict its overall behavior, identify potential maxima and minima, and solve optimization problems effectively.

Related terms

Increasing Function: An increasing function is a function where, as the input values increase, the output values also increase.

Local Maximum: A local maximum is a point on the graph of a function where the output value is higher than all nearby output values.

Critical Point: A critical point is a point on the graph of a function where the derivative is zero or undefined, often indicating a local maximum, local minimum, or inflection point.