Question

По стройте график функции y=1/2( одна вторая если что) на промежутке(-∞; 0). А) вазрастаюшей или убывающей является данная функция на этом промежутке? Б) с помощью определения докажите свое утверждение в пункте а.

Answer 1

1)y=1/x,x∈(0;∞)
x 1/2  1      2
y  2    1      1/2
Функция убывающая
x1<x2    x1=1/2  x2=1
y1>y2    y1=2  y2=1
2)

Answer 2

Graph of the Function y = 1/2x on the Interval (-∞, 0)

To plot the graph of the function y = 1/2x on the interval (-∞, 0), we can start by selecting some x-values within this interval and calculating the corresponding y-values. Let's choose a few x-values and calculate their corresponding y-values:

| x | y = 1/2x | | --- | -------- | | -2 | -1 | | -1 | -1/2 | | -0.5| -1/4 | | -0.1| -1/20 | | -0.01| -1/200 | | -0.001| -1/2000|

Now, let's plot these points on a graph:


As we can see from the graph, the function y = 1/2x is decreasing on the interval (-∞, 0). This means that as x approaches 0 from the left side, the corresponding y-values decrease.

Proof of the Function's Decreasing Nature

To prove that the function y = 1/2x is decreasing on the interval (-∞, 0), we can use the definition of a decreasing function. According to the definition, a function f(x) is decreasing on an interval if for any two values a and b in the interval, where a < b, f(a) > f(b).

Let's consider two values, a and b, in the interval (-∞, 0) where a < b. We will show that f(a) > f(b) for the function y = 1/2x.

Let a = -2 and b = -1. Since a < b, we can compare f(a) and f(b):

f(a) = 1/2(-2) = -1 f(b) = 1/2(-1) = -1/2

As we can see, f(a) = -1 is greater than f(b) = -1/2. This confirms that the function y = 1/2x is decreasing on the interval (-∞, 0).

Therefore, we can conclude that the function y = 1/2x is a decreasing function on the interval (-∞, 0).