Question

Y=ln cosx четная или нечетная?

Answer 1

Y(-x) = ln(cos(-x)) = ln(cos(x)) = y(x) - функция четная

Answer 2

To determine if the function Y = ln(cos(x)) is even or odd, we need to understand the properties of even and odd functions.

1. Even Function: A function f(x) is even if and only if f(-x) = f(x) for all x in its domain. In other words, the function is symmetric about the y-axis.

2. Odd Function: A function f(x) is odd if and only if f(-x) = -f(x) for all x in its domain. In other words, the function has rotational symmetry about the origin.

Now, let's check if Y = ln(cos(x)) satisfies the conditions for even or odd functions:

1. Even property: Y(-x) = ln(cos(-x))

Now, since cos(-x) = cos(x), we get: Y(-x) = ln(cos(x))

However, ln(cos(x)) is NOT equal to Y(x) = ln(cos(x)) because the natural logarithm function (ln) is not symmetric about the y-axis. Therefore, Y = ln(cos(x)) is NOT an even function.

2. Odd property: Y(-x) = ln(cos(-x))

As we established earlier, cos(-x) = cos(x), so: Y(-x) = ln(cos(x))

Now, we need to check if -Y(x) = ln(cos(x)): -Y(x) = -ln(cos(x))

Since -ln(cos(x)) is indeed equal to Y(-x) = ln(cos(x)), we can conclude that Y = ln(cos(x)) is an odd function.

In summary, the function Y = ln(cos(x)) is an odd function.