are these functions periodic?

i - y=abs(sin(x)) = sin(x)
ii - y=cos(X^2)
iii - y = cos(sin(x))

I am completely lost on these :/

yes

no , http://www.wolframalpha.com/input/?i=cos%28x%5E2%29

yes , http://www.wolframalpha.com/input/?i=cos%28sin%28x%29%29

To determine if a function is periodic or not, you need to check if it repeats itself after a certain interval. For periodic functions, there exists a positive number called the period, which represents the interval after which the function repeats.

i) Let's analyze the first function: y = |sin(x)|.

To determine if this function is periodic, we need to examine the behavior of sin(x). The sine function, sin(x), is periodic with a period of 2π. However, when we take the absolute value of sin(x), i.e., |sin(x)|, the negative part of the wave, below the x-axis, becomes positive, resulting in a wave that has no negative values. Hence, the absolute value function does not have a period. Therefore, y = |sin(x)| is not a periodic function.

ii) Moving on to the second function: y = cos(X^2).

Examining this equation, we can see that the input of the cosine function is x^2. Squaring a value only increases it positively or keeps it at zero, meaning the values of x^2 are always non-negative or zero.

Since cosine oscillates between -1 to 1, it becomes evident that the cosine function applied to x^2 will still oscillate between -1 to 1 but with higher frequency compared to the original cosine function.

In other words, the graph of cos(X^2) is squeezing the usual cosine wave horizontally. Consequently, the graph will not repeat itself over any fixed interval. Thus, y = cos(X^2) is not a periodic function.

iii) Lastly, let's address the function: y = cos(sin(x)).

In this case, we are taking the sine of x and then applying the cosine to it. As we mentioned earlier, the sine function has a period of 2π. When we take the cosine of sine, the resulting function behaves similarly, oscillating between -1 to 1 but with a higher frequency.

Since both the sine and cosine functions are periodic with the same period, 2π, the composition of these functions, y = cos(sin(x)), is also periodic with a period of 2π. Therefore, y = cos(sin(x)) is a periodic function.

In summary:
- The function y = |sin(x)| is not periodic.
- The function y = cos(X^2) is not periodic.
- The function y = cos(sin(x)) is periodic with a period of 2π.