A strictly decreasing function is a type of function where, for any two points in its domain, if the first point is less than the second point, then the value of the function at the first point is greater than the value of the function at the second point. This means that as you move along the x-axis from left to right, the y-values consistently decrease. Such functions play a crucial role in understanding monotonicity, which relates to how functions behave in terms of increasing and decreasing over their domains.
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For a function to be strictly decreasing, it must hold true that for all x1 and x2 in its domain, if x1 < x2 then f(x1) > f(x2).
Graphically, a strictly decreasing function will slope downwards as you move from left to right across the graph.
Strictly decreasing functions do not allow for constant segments; every segment must show a decrease.
Examples of strictly decreasing functions include f(x) = -x and f(x) = -x^2 (for x < 0).
Strictly decreasing functions are useful for finding intervals where a function can be inverted since they are one-to-one.
Review Questions
How can you determine if a given function is strictly decreasing over an interval?
To determine if a function is strictly decreasing over an interval, you need to analyze its derivative. If the derivative of the function is negative for all points in that interval, then the function is strictly decreasing. Additionally, you can select two points within the interval; if for any two points x1 and x2, where x1 < x2, it holds that f(x1) > f(x2), then this confirms that the function is strictly decreasing.
Discuss the implications of having a strictly decreasing function when finding its inverse.
When a function is strictly decreasing, it means that it is one-to-one, allowing us to find an inverse that is also a function. This is important because inverses only exist for functions that pass the horizontal line test. The fact that there are no repeated y-values guarantees that each output corresponds to exactly one input, making it possible to define an inverse over its entire range without ambiguity.
In optimization problems involving strictly decreasing functions, strict monotonicity simplifies finding maximum or minimum values. Since these functions continuously decline without leveling off, they will not have local maxima within their domain. Instead, their global maximum occurs at the boundary of their defined interval. For instance, if we want to maximize profit modeled by a strictly decreasing revenue function, we know that maximizing occurs at the highest end of our input variable range. This understanding directly influences strategy decisions based on how variables interact within these functions.
A function that is either entirely non-increasing or non-decreasing throughout its domain.
increasing function: A function where, for any two points in its domain, if the first point is less than the second point, then the value of the function at the first point is less than or equal to the value at the second point.
local maximum: A point in a function where the value of the function is greater than the values immediately adjacent to it, which helps in identifying peaks in a graph.