Question

Исследовать функцию и построить ее график : y= x/x²-4

Answer 1

ДАНО

Y= x/(x²-4)

ИССЛЕДОВАНИЕ

1.Область определения D(x). В знаменателе не ноль.  (x²-4)=(x-2)(x+2)≠0

 Х∈(-∞;-2)∪(-2;2)∪(2;+∞).

Вертикальных асимптоты (две) - х1 = -2,.х2 = 2.

2. Пересечение с осью Х. Y=0 при х = 0. 

3. Пересечение с осью У.  У(0) = 0. 

4. Поведение на бесконечности.(Делим и числитель и знаменатель на Х  в степени числителя)

$Y= \frac{1}{x- \frac{4}{x} }=0$  

limY(+∞) = 0. 

Горизонтальная асимптота   Y=0.

5. Исследование на чётность.Y(-x) = - Y(x).

Функция нечётная. 

6. Производная функции.

$Y'(x)= \frac{1}{(x^2-4)}- \frac{2x^2}{(x^2-4)^2}$

 Корней  - нет.

7. Локальные экстремумы.Максимума и минимума  – нет.

8. Интервалы монотонности. 

Убывает на всем интервале определения.

9. Вторая производная - Y"(x).

 $Y"(x)= \frac{-6x}{(x^2-4)^2}+ \frac{8x^3}{(x^2-4)^3}$

Корни производной - точки перегиба:  х = 0 и в точках разрыва вычисляем пределы при х = +/-2. 

9. Выпуклая “горка» Х∈(-∞;-2)∪(0;2), Вогнутая – «ложка» Х∈(-2;0)∪(2;+∞). 

10. Область значений Е(у) У∈(-∞;+∞) 

11. Наклонная асимптота. Уравнение: lim(∞)(k*x+b – f(x).  

k=lim(∞)Y(x)/x = 1/(x²-4) = 0. Совпадает с горизонтальной асимптотой. 

12.График в приложении.

Answer 2

Function Analysis and Graph

The given function is y = x / (x² - 4).

To analyze the function and plot its graph, we can start by examining its domain and any vertical asymptotes.

The denominator of the function, x² - 4, cannot be equal to zero, as division by zero is undefined. Therefore, we need to find the values of x that make the denominator zero and exclude them from the domain.

Solving x² - 4 = 0, we get x = ±2. So, the function is undefined at x = -2 and x = 2.

Next, let's analyze the behavior of the function as x approaches positive and negative infinity.

As x approaches positive infinity, both the numerator (x) and the denominator (x² - 4) grow without bound. Therefore, the function approaches zero.

As x approaches negative infinity, again, both the numerator (x) and the denominator (x² - 4) grow without bound. Therefore, the function approaches zero.

Hence, the horizontal asymptote of the function is y = 0.

Now, let's find the x-intercepts and y-intercepts of the function.

To find the x-intercepts, we set y = 0 and solve for x:

0 = x / (x² - 4)

Since the numerator is zero, the only solution is x = 0. Therefore, the x-intercept is (0, 0).

To find the y-intercept, we set x = 0 and solve for y:

y = 0 / (0² - 4) = 0 / (-4) = 0

Therefore, the y-intercept is (0, 0).

Now, let's plot the graph of the function.

The graph will have a vertical asymptote at x = -2 and x = 2, a horizontal asymptote at y = 0, and a point at the origin (0, 0).


Please note that the graph is a rough representation and may not be perfectly accurate. It is always recommended to use graphing software or calculators for precise graphs.

I hope this helps! Let me know if you have any further questions.