Question

Указать область определения функции: y=3x+1/4x^2-11x-3

Answer 1

Решение задания смотри на фотографии

Answer 2

To find the domain of a function, we need to identify all the values of "x" for which the function is defined. The domain consists of all the possible input values that produce a valid output.

In this case, the function is given by:

y = (3x + 1) / (4x^2 - 11x - 3)

To determine the domain, we need to consider two factors:

1. Division by zero: The function is undefined when the denominator (4x^2 - 11x - 3) becomes zero, as division by zero is not allowed.

2. Square root of a negative number (for real numbers): The function may involve square roots in certain cases, and for the function to be defined for real numbers, the expression inside the square root must be non-negative.

Now, let's find the domain step by step:

Step 1: Find the values of "x" that make the denominator zero.

4x^2 - 11x - 3 = 0

We can solve this quadratic equation by factoring or using the quadratic formula:

Using the quadratic formula: x = [11 ± √(11^2 - 4 * 4 * -3)] / 2 * 4 x = [11 ± √(121 + 48)] / 8 x = [11 ± √169] / 8 x = [11 ± 13] / 8

So, we have two possible values for "x":

x = (11 + 13) / 8 = 24 / 8 = 3 x = (11 - 13) / 8 = -2 / 8 = -1/4

Step 2: Combine the results from Step 1 with any other restrictions, if present.

The function is undefined for x = 3 (division by zero), and it is also undefined for x = -1/4 (division by zero).

So, the domain of the function is all real numbers except x = 3 and x = -1/4.

In interval notation, the domain is (-∞, -1/4) ∪ (-1/4, 3) ∪ (3, ∞).