Question

Найти область определения функции z=arcsin(2x-y)

Answer 1

Решение : ///////////////////////////////////////

Answer 2

To find the domain of the function $z = \arcsin(2x-y)$, we need to determine the set of all possible values for the independent variables $x$ and $y$ that make the function valid.

The function $z = \arcsin(2x-y)$ represents the inverse sine function applied to the expression $2x-y$. The domain of the arcsine function is limited by the range of the regular sine function, which is between -1 and 1. So, to find the domain, we need to find the values of $x$ and $y$ that make $2x-y$ fall within this range.

Mathematically, we have the following inequality: $-1 \leq 2x - y \leq 1$

To simplify the inequality, let's split it into two separate inequalities: $-1 \leq 2x - y$ $2x - y \leq 1$

Now, we solve each inequality for $x$ and $y$ separately.

1. For $-1 \leq 2x - y$, solving for $y$: $y \leq 2x + 1$

2. For $2x - y \leq 1$, solving for $y$: $y \geq 2x - 1$

Now, we need to find the common solution for both inequalities. Since the domain of $x$ and $y$ must satisfy both conditions, the domain is the intersection of the two regions:

$y \leq 2x + 1 \quad \text{and} \quad y \geq 2x - 1$

This means that the domain of the function $z = \arcsin(2x-y)$ is the region between the lines $y = 2x + 1$ and $y = 2x - 1$ in the $xy$-plane.

Graphically, this domain corresponds to the area between the two parallel lines. Any point within this region will produce a real value for $\arcsin(2x-y)$.