Increasing and Decreasing Functions
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Hi,
Different textbooks have different ways of defining increasing or decreasing on an interval.
The most common calculus textbook currently used in US universities is James Stewart's "Calculus: Early Transcendentals" (9th Edition).
In this particular textbook, a function is defined to be increasing on an interval I if, for every pair of numbers in that interval, a and b, with a < b, it is the case that f(a) < f(b) (defined on page 17 of the textbook referenced above).
In other words, f is increasing on an interval if the output value of f increases as the input value of f increases.
A function is defined to be decreasing on an interval I similarly: f is decreasing on an interval I if, for every pair of numbers in that interval, a and b, with a < b, it is the case that f(a) > f(b).
In other words, f is decreasing on an interval if the output value of f decreases as the input value increases.
A discrepancy arises later on in the textbook with the definition of a function increasing at a point (which is slightly different from increasing on an interval).
Later on, after derivatives have been defined, it is demonstrated that you can detect an interval on which a function is increasing by simply looking at the sign of the derivative of the function on that interval (instead of comparing values at every pair of points within the interval).
If the sign of the derivative is positive at every point within an interval, it can be shown that the function is increasing on that interval (similarly with decreasing and a negative derivative).
This leads to the idea of a function increasing at a point if the function's derivative is positive at that point.
Herein lies the problem: this detection mechanism has a discrepancy at the endpoints of the interval of increase or decrease: usually, at the endpoint of an interval where the function is increasing, the derivative is either 0 or undefined (such points are called critical points).
This leads to a discrepancy: in the original definition of increasing on an interval, the endpoint would be included, as it fits the definition, as long as the function is defined and continuous there. However, the derivative either says nothing about the endpoint's behavior (if the derivative is undefined), or it has a value of 0 (which is associated with stationary behavior if it occurs on an interval).
As such, students are sometimes confused about whether to include the endpoints when asked about intervals of increase or decrease.
From the original definition, as long as the endpoint fits the definition, it can be included. However, many teachers also accept exclusion of the endpoints, even if they fit the original definition (this discrepancy is covered on pages 297-298 of Stewart's Calculus: Early Transcendentals (9th Edition), with the consensus that both answers should be accepted).
If you do not use this textbook, you should consult your notes or your textbook to see what definition is being used, as the definition differs considerably (some textbooks use ≤ and ≥ instead of < and >, for example).
That being said, on the lower right corner of the graph in your shared picture, there is an icon of an idealized magnifying glass. Clicking on that icon should allow you to zoom in on the turning points of the graph (where the graph changes from increasing to decreasing or vice versa), and thus more clearly determine the x-coordinates of those points.
Without an equation or formula defining the curve, there is no way to determine the exact x-coordinates of the turning points. However, you are probably meant to assume that the x-coordinates of the turning points are either integers or common fractions (fractions with relatively small numerators and denominators).
As such, let us examine the behavior of the y-coordinates of the graph from left to right, as that is the direction of increasing x-coordinates.
From the left up to what appears to be x = -3, the y-coordinates of the graph decrease. When we cannot see the leftmost endpoint of a graph like this one, the author most likely wants us to assume that the behavior of the graph, in this case decreasing y-coordinates, continues indefinitely to the left.
Thus, we may conclude that the graph decreases all the way from -∞ up to -3, which would be written in interval notation as (-∞, -3] if we include the endpoint or (-∞, -3) if we exclude it, both of which are correct, as noted in the discussion above. Note that -∞ and ∞ are never included because they are not in the domain of the function; they just formally denote that the interval has no endpoint in that direction.
Continuing rightwards, from -3 to what appears to be halfway between -1 and 0, the y-coordinates increase. We would therefore say the graph is increasing on the interval [-3, -½] or (-3, -½).
From x = -½ rightwards, the y-coordinates of the graph appear to decrease, so we would say that the graph is decreasing on the interval [-½, ∞) or (-½, ∞).
Thus, using the union symbol ∪ to unite disjoint intervals, we would say that the graphed function decreases over the interval (-∞, -3] ∪ [-½, ∞) or the interval (-∞, -3) ∪ (-½, ∞) and it increases over the interval [-3, -½] or (-3, -½), depending on the conventions adopted by your teacher.
Different textbooks have different ways of defining increasing or decreasing on an interval.
The most common calculus textbook currently used in US universities is James Stewart's "Calculus: Early Transcendentals" (9th Edition).
In this particular textbook, a function is defined to be increasing on an interval I if, for every pair of numbers in that interval, a and b, with a < b, it is the case that f(a) < f(b) (defined on page 17 of the textbook referenced above).
In other words, f is increasing on an interval if the output value of f increases as the input value of f increases.
A function is defined to be decreasing on an interval I similarly: f is decreasing on an interval I if, for every pair of numbers in that interval, a and b, with a < b, it is the case that f(a) > f(b).
In other words, f is decreasing on an interval if the output value of f decreases as the input value increases.
A discrepancy arises later on in the textbook with the definition of a function increasing at a point (which is slightly different from increasing on an interval).
Later on, after derivatives have been defined, it is demonstrated that you can detect an interval on which a function is increasing by simply looking at the sign of the derivative of the function on that interval (instead of comparing values at every pair of points within the interval).
If the sign of the derivative is positive at every point within an interval, it can be shown that the function is increasing on that interval (similarly with decreasing and a negative derivative).
This leads to the idea of a function increasing at a point if the function's derivative is positive at that point.
Herein lies the problem: this detection mechanism has a discrepancy at the endpoints of the interval of increase or decrease: usually, at the endpoint of an interval where the function is increasing, the derivative is either 0 or undefined (such points are called critical points).
This leads to a discrepancy: in the original definition of increasing on an interval, the endpoint would be included, as it fits the definition, as long as the function is defined and continuous there. However, the derivative either says nothing about the endpoint's behavior (if the derivative is undefined), or it has a value of 0 (which is associated with stationary behavior if it occurs on an interval).
As such, students are sometimes confused about whether to include the endpoints when asked about intervals of increase or decrease.
From the original definition, as long as the endpoint fits the definition, it can be included. However, many teachers also accept exclusion of the endpoints, even if they fit the original definition (this discrepancy is covered on pages 297-298 of Stewart's Calculus: Early Transcendentals (9th Edition), with the consensus that both answers should be accepted).
If you do not use this textbook, you should consult your notes or your textbook to see what definition is being used, as the definition differs considerably (some textbooks use ≤ and ≥ instead of < and >, for example).
That being said, on the lower right corner of the graph in your shared picture, there is an icon of an idealized magnifying glass. Clicking on that icon should allow you to zoom in on the turning points of the graph (where the graph changes from increasing to decreasing or vice versa), and thus more clearly determine the x-coordinates of those points.
Without an equation or formula defining the curve, there is no way to determine the exact x-coordinates of the turning points. However, you are probably meant to assume that the x-coordinates of the turning points are either integers or common fractions (fractions with relatively small numerators and denominators).
As such, let us examine the behavior of the y-coordinates of the graph from left to right, as that is the direction of increasing x-coordinates.
From the left up to what appears to be x = -3, the y-coordinates of the graph decrease. When we cannot see the leftmost endpoint of a graph like this one, the author most likely wants us to assume that the behavior of the graph, in this case decreasing y-coordinates, continues indefinitely to the left.
Thus, we may conclude that the graph decreases all the way from -∞ up to -3, which would be written in interval notation as (-∞, -3] if we include the endpoint or (-∞, -3) if we exclude it, both of which are correct, as noted in the discussion above. Note that -∞ and ∞ are never included because they are not in the domain of the function; they just formally denote that the interval has no endpoint in that direction.
Continuing rightwards, from -3 to what appears to be halfway between -1 and 0, the y-coordinates increase. We would therefore say the graph is increasing on the interval [-3, -½] or (-3, -½).
From x = -½ rightwards, the y-coordinates of the graph appear to decrease, so we would say that the graph is decreasing on the interval [-½, ∞) or (-½, ∞).
Thus, using the union symbol ∪ to unite disjoint intervals, we would say that the graphed function decreases over the interval (-∞, -3] ∪ [-½, ∞) or the interval (-∞, -3) ∪ (-½, ∞) and it increases over the interval [-3, -½] or (-3, -½), depending on the conventions adopted by your teacher.
