Question

Надие область определения функции y=3x/x^2-3x-4

Answer 1

Решение задания смотри на фотографии

Answer 2

To find the domain of the function $y = \frac{3x}{x^2 - 3x - 4}$, we need to identify the values of $x$ for which the function is defined.

The domain of a rational function like this one consists of all real numbers except the values that would cause the denominator to become zero. Division by zero is undefined in mathematics, so we must avoid these values.

In this case, the denominator is $x^2 - 3x - 4$. To find the values that make the denominator zero, we can set it equal to zero and solve for $x$:

$x^2 - 3x - 4 = 0$

To factor this quadratic equation, we need to find two numbers whose product is $ac$ (in this case $1 \times -4 = -4$) and whose sum is $b$ (in this case $-3$).

The two numbers that fit these criteria are $4$ and $-1$ because $4 \times -1 = -4$ and $4 + (-1) = -3$.

Now we can rewrite the quadratic equation as:

$x^2 - 4x + x - 4 = 0$

Factor by grouping:

$x(x - 4) + 1(x - 4) = 0$

Now, we can factor out the common factor $(x - 4)$:

$(x - 4)(x + 1) = 0$

Now set each factor to zero and solve for $x$:

$x - 4 = 0 \quad \Rightarrow \quad x = 4$

$x + 1 = 0 \quad \Rightarrow \quad x = -1$

The values $x = 4$ and $x = -1$ make the denominator zero. Therefore, they are not included in the domain.

The domain of the function $y = \frac{3x}{x^2 - 3x - 4}$ is all real numbers except $x = 4$ and $x = -1$. In interval notation, the domain can be expressed as:

Domain: $(- \infty, -1) \cup (-1, 4) \cup (4, + \infty)$