Question

Найти область определения функции: y=log2(x-x^2).

Answer 1

Ответ:(0;1)

Объяснение:

х-х²>0

х(1-х) >0

Найдем точки х=0 и х =1. Через эти точки проходит парабола ветви вниз. Нам нужен знаю >0, тогда область определения х€(0;1)

Answer 2

To find the domain of the function $y = \log_2(x - x^2)$, we need to identify the values of $x$ for which the function is defined. The domain of a logarithmic function is determined by the argument of the logarithm, and for $\log_b(a)$ to be defined, $a$ must be greater than zero.

In this case, we have the function $y = \log_2(x - x^2)$. For the logarithm to be defined, the argument $(x - x^2)$ must be greater than zero. Therefore, we need to solve the inequality:

$x - x^2 > 0$

To solve this inequality, follow these steps:

Step 1: Factor out $x$ from the expression:

$x(1 - x) > 0$

Step 2: Find the critical points by setting each factor equal to zero and solving for $x$:

$x = 0$ and $1 - x = 0$

$x = 0$ and $x = 1$

Step 3: Create a sign chart with these critical points and test points within each interval:

$$\begin{array}{|c|c|c|c|c|} \hline & x < 0 & 0 < x < 1 & x > 1 \\ \hline x & -1 & \frac{1}{2} & 2 \\ \hline x - x^2 & - & + & + \\ \hline \end{array}$$

Step 4: Based on the sign chart, we can see that the inequality is satisfied when $x < 0$ or $0 < x < 1$. Therefore, the domain of the function $y = \log_2(x - x^2)$ is the set of all real numbers $x$ that fall within these intervals:

$\text{Domain:} \quad (-\infty, 0) \cup (0, 1)$

Please note that the domain does not include the point $x = 1$ because $x - x^2 = 1 - 1^2 = 0$, and the logarithm of zero is undefined.