Lecture Note
University
Duke UniversityCourse
Data Science Math SkillsPages
4
Academic year
2023
KatrCrayon
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Functions - Increasing and Decreasing Functions Increasing and decreasing functions are two special classes of functions; mostfunctions do not fall into either category but it is important to understand which oneeach function falls into. When we look to the left, we see three functions: the graph of f of x, which is the redgraph; g of x, which is the blue graph; and h of x, which is the yellow graph. You'llnotice that as one moves along the x axis and looks at where one is on a particulargraph, one climbs up (positive) on f of x but falls on g of x. The graph of g is always falling as I move along the x-axis. The graph of h has afalling period and then an increasing period. We say that f is a strictly increasingfunction. We say that g is a strictly decreasing function. And it turns out that h isneither. As we often do as mathematicians, we are not satisfied with just visualdefinitions. On the right, we've plotted several functions, with their derivativesindicated by blue and red arrows. Let's take a look at what these variables represent,and how we can derive the intuitive definitions of these functions from the symbolschematics. Recall that the red arrows indicate increasing values, while blue arrowsindicate decreasing values; h indicates that a variable is neither increasing nordecreasing. If f is a function from the real line to the real line, and f can stand in for any of thesefunctions, we say that f is strictly increasing if—a condition that may be difficult tounderstand. Whenever two inputs, a and b, have a relationship that a is less than b,it must be that the output f_of_a, f_of_b also have this same relationship. Let's check that f is strictly increasing. For example, if this is the point a and this is the point b,notice that a and b are less, a is less than b. So look up here, there is f_of_a on thegraph, there's f_of_b. And notice that this relationship is the same for both functions.This means that the function f is strictly increasing. We say that f is strictlydecreasing if whenever a is less than b, the order of relationship flips the output, sof_of_a is greater than f_of_b. The blue line that you see here is actually true, but ifyou look at the blue line, you will notice that g_of_b is less than g_of_a. The twofunctions are getting closer together and g_of_b is always going to be less thang_of_a no matter where we pick a and b. Now we can find the value of h. First, notethat the function g is strictly decreasing. Second, since g is strictly decreasing and ifwe take a and b, where a < b and g(a) > g(b), then h(a) > h(b). On the other hand, if we substitute a and b with the numbers we have here and lookat where each value hits the graph, h_of_a is actually less than h_of_b. Therefore,there's no consistency in our values. Let's take a look at some examples of what thismight mean in practice.
So when we consider the function f(x) = 2 to the x. Then we considered g(x) = 3 tothe minus x, and one you've seen before which is h(x) = x squared. Now let's figureout which of these are strictly increasing, which are strictly decreasing, which areneither. Over here, our axis. And let's first start by drawing the graph of f(x) = 2x tothe X. This is called an exponential function. The variable x is in the x-position. Let'schart out our table of values and see what that looks like. If x is zero, this is x and this is f(x), if x is 0, then we have two to the 0, which is 1.And here's the 0.01. X is 1, we have 2 to the 1 equals 2; x is 2, we have 2 cubed, or8. Shoots all the way up there. X is negative 1: 2 to the minus 1 is 1 over 2 to the 1,which is 1/2; and so on. The graph of f(x) = 2 to the x is asymptotic to the x-axis,which means that it approaches but never reaches x = 0. The graph is strictlyincreasing, which means that its slope (gradient) is positive for all values of x.Okay, let's figure out the graph of g(x) = 3 to the minus x. We can make a table ofvalues to come up with an equation for the line. Here we have x and g(x). The value0 is included in the table since it is a point on the line. We see that g(0) = 1.Therefore, when x = 0, g(x) = 1. The pattern here is going to go steeper and thenflatter like so (more steeply initially). Therefore, we can state that g(x) is strictlydecreasing over its domain. Let's draw the graph of h(x) = x2 in yellow. So let's see that it should go through thepoint (1,1) and (-1,1). Now h is not strictly increasing or decreasing on the entire realline, but only on the interval from zero to infinity. In other words, if we restrict
ourselves only to points in this interval here and only use those as inputs, h satisfiesthe definition of strictly increasing. We plug in two points in that interval: it goes upand h is strictly decreasing on the interval from minus infinity to zero. Let's take a look at some examples of functions that might be more applicable to thereal world, such as those that relate to the development of children. On the left, wecan plot years since birth against height. On the other hand, if we look at negativeyears since birth, there's not much point in considering any values to the left of zerobecause they aren't negative numbers. This will be an increasing function almostentirely. Right, you'll start at year one, we'll measure you about here, and you'll goup, you go up really violently and then just level off. Right around here that'sprobably about 17, you'll stay stable for a long time perhaps with a bit of growth, andthen sadly decline if things don't improve. So, that's a function which is increasing fora long bit, then it's sort of flat and then goes down. As always the mathematicalnotions will be much more precise and what happens in real life, wouldn't be asmooth curve, it would depend on the measurements of the doctor's office. Supposewe plot the year of purchase on the x-axis and the value of a car on the y-axis.Without even drawing it, you probably predict that there will be a decreasing function.Here is year one, it's one of my cars, it starts about a thousand dollars. Perhapsthat's revaluing me, and then it will decrease. As soon as you drive it off the lot ofgoes down a little bit, and it will plateau somewhere near the bottom.
We're going to end with a simple visual test to determine whether a function isincreasing or decreasing. The red graph represents a strictly increasing function. Theblue graph indicates that the function is not strictly increasing or decreasing. Thehorizontal line test is a simple way to do this. Drawing a line to the right-hand graphwill always hit a value once, regardless of where on the line it is drawn. This makessense, because that value is the only one hit by that line. If the line were movedslightly higher or lower, it would not return to this value because if it did - the graphwould have to bend back down. In other words, there is only one point on the linethat hits a specific value and no other point on the line could ever reach that samevalue because the curve has ended. On the other hand, you will notice that blue hasmany horizontal lines that intersect it twice. This one hit it here, then went down andhit it again, and again, and again. So blue goes down forever, but then comes backup and hits the graph again and again. This is called the horizontal line test for strictmonotonicity. If a function is strictly increasing or strictly decreasing whenever yougraph it, every horizontal line must intersect it exactly once.
Functions - Increasing and Decreasing Functions
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