We first met in a 7th grade algebra course. Looking at the graph of the function, we took down the corresponding information: if, moving along the graph from left to right, we at the same time move from bottom to top (as if climbing a hill), then we declared the function to be increasing (Fig. 124); if we move from top to bottom (go down a hill), then we declared the function to be decreasing (Fig. 125).
However, mathematicians are not very fond of this method of studying the properties of a function. They believe that definitions of concepts should not be based on a drawing - the drawing should only illustrate one or another property of a function on its graphics. Let us give strict definitions of the concepts of increasing and decreasing functions.
Definition 1.
The function y = f(x) is said to be increasing on the interval X if, from the inequality x 1< х 2 - где хг и х2 - любые две точки промежутка X, следует неравенство f(x 1) < f(x 2).
Definition 2. The function y = f(x) is said to be decreasing on the interval X if the inequality x 1< х 2 , где х 1 и х 2 - любые две точки промежутка X, следует inequality f(x 1) > f(x 2).
In practice, it is more convenient to use the following formulations:
a function increases if a larger value of the argument corresponds to a larger value of the function;
a function decreases if a larger value of the argument corresponds to a smaller value of the function.
Using these definitions and the properties established in § 33 numerical inequalities, we will be able to justify conclusions about the increase or decrease of previously studied functions.
1. Linear function y = kx +m
If k > 0, then the function increases throughout (Fig. 126); if k< 0, то функция убывает на всей числовой прямой (рис. 127).
Proof. Let f(x) = kx +m. If x 1< х 2 и k >Oh, then, according to the property of 3 numerical inequalities (see § 33), kx 1< kx 2 . Далее, согласно свойству 2, из kx 1 < kx 2 следует, что kx 1 + m < kx 2 + m, т. е. f(х 1) < f(х 2).
So, from the inequality x 1< х 2 следует, что f(х 1) < f(x 2). Это и означает возрастание функции у = f(х), т.е. linear functions y = kx+ m.
If x 1< х 2 и k < 0, то, согласно свойству 3 числовых неравенств, kx 1 >kx 2 , and according to property 2, from kx 1 > kx 2 it follows that kx 1 + m> kx 2 + i.e.
So, from the inequality x 1< х 2 следует, что f(х 1) >f(x 2). This means a decrease in the function y = f(x), i.e. linear function y = kx + m.
If a function increases (decreases) throughout its entire domain of definition, then it can be called increasing (decreasing) without indicating the interval. For example, about the function y = 2x - 3 we can say that it is increasing along the entire number line, but we can also say it more briefly: y = 2x - 3 - increasing
function.
2. Function y = x2
1. Consider the function y = x 2 on the ray. Let's take two non-positive numbers x 1 and x 2 such that x 1< х 2 . Тогда, согласно свойству 3 числовых неравенств, выполняется неравенство - х 1 >- x 2. Since the numbers - x 1 and - x 2 are non-negative, then by squaring both sides of the last inequality, we obtain an inequality of the same meaning (-x 1) 2 > (-x 2) 2, i.e. This means that f(x 1) > f(x 2).
So, from the inequality x 1< х 2 следует, что f(х 1) >f(x 2).
Therefore, the function y = x 2 decreases on the ray (- 00, 0] (Fig. 128).
1. Consider a function on the interval (0, + 00).
Let x1< х 2 . Так как х 1 и х 2 - , то из х 1 < x 2 следует (см. пример 1 из § 33), т. е. f(x 1) >f(x 2).
So, from the inequality x 1< х 2 следует, что f(x 1) >f(x 2). This means that the function decreases on the open ray (0, + 00) (Fig. 129).
2. Consider a function on the interval (-oo, 0). Let x 1< х 2 , х 1 и х 2 - negative numbers. Then - x 1 > - x 2, and both sides of the last inequality are positive numbers, and therefore (we again used the inequality proven in example 1 from § 33). Next we have, where we get from.
So, from the inequality x 1< х 2 следует, что f(x 1) >f(x 2) i.e. function decreases on the open ray (- 00 , 0)
Usually the terms “increasing function” and “decreasing function” are combined under the general name monotonic function, and the study of a function for increasing and decreasing is called the study of a function for monotonicity.
Solution.
1) Let’s plot the function y = 2x2 and take the branch of this parabola at x< 0 (рис. 130).
2) Construct and select its part on the segment (Fig. 131).
3) Let's construct a hyperbola and select its part on the open ray (4, + 00) (Fig. 132).
4) Let us depict all three “pieces” in one coordinate system - this is the graph of the function y = f(x) (Fig. 133).
Let's read the graph of the function y = f(x).
1. The domain of definition of the function is the entire number line.
2. y = 0 at x = 0; y > 0 for x > 0.
3. The function decreases on the ray (-oo, 0], increases on the segment, decreases on the ray, is convex upward on the segment, convex downward on the ray)