Question

Найдите область определения функции: y=1/cos(2x-3)

Answer 1

Знаменатель дроби не может быть равен 0, поэтому
cos(2x-3) ≠ 0
2x - 3 ≠ π/2 + π*n, где n - целое
x ≠ π/4 + π*n/2 + 3/2

Answer 2

To find the domain of the function y = 1/cos(2x - 3), we need to consider the values of x for which the expression is defined. The cosine function has a domain of all real numbers, but we need to be mindful of the argument inside the cosine function, which is 2x - 3.

The domain of the function y = 1/cos(2x - 3) is determined by the domain of the inner function, which in this case is 2x - 3. To ensure that the inner function is valid, we look for the following two conditions:

1. The denominator of the fraction should not be equal to zero, as division by zero is undefined.
2. The expression inside the cosine function should be in the valid domain of the cosine function.

Condition 1: The denominator of the fraction is cos(2x - 3). To avoid division by zero, cos(2x - 3) should not be equal to zero:

cos(2x - 3) ≠ 0

Condition 2: The expression inside the cosine function should be within the valid domain of the cosine function. The cosine function is defined for all real numbers, so there are no additional restrictions from the cosine function itself.

Now, we will solve for the values of x that satisfy Condition 1:

cos(2x - 3) ≠ 0

To find the solutions for this inequality, we need to find the values of x that make the cosine function equal to zero. The cosine function is zero at certain specific angles:

cos(θ) = 0 when θ = (2n + 1)π/2, where n is an integer.

Let's apply this to our case where θ = 2x - 3:

2x - 3 = (2n + 1)π/2

Now, solve for x:

2x = (2n + 1)π/2 + 3

x = [(2n + 1)π/2 + 3] / 2

The expression [(2n + 1)π/2 + 3] / 2 gives us the values of x at which the denominator cos(2x - 3) becomes zero. However, we also need to consider the restriction on the domain of the inner function 2x - 3.

Since the cosine function has a periodic nature, the solutions for x may repeat after each period of the cosine function (which is 2π). Therefore, we can express the general solution for the domain of x as follows:

x = [(2n + 1)π/2 + 3] / 2, where n is an integer.

So, the domain of the function y = 1/cos(2x - 3) is all real numbers except those values of x that satisfy the above equation.