Determine whether the function is periodic. State the period, the domain and the range of g(x)= 2 cos 2x+3 sin 5x

To determine if the function g(x) = 2cos(2x) + 3sin(5x) is periodic, we need to check whether there exists a positive number T such that g(x + T) = g(x) for all values of x. Let's break it down step by step:

Step 1: Finding the Period of the Function
The period of a trigonometric function is the smallest positive value T for which the function repeats itself. For individual cosine and sine functions, their periods are 2π and 2π/5, respectively. To find the period of g(x), we need to find the least common multiple (LCM) of the individual periods, which will give us the period of the combined function.

LCM(2π, 2π/5)
Convert both fractions to have the same denominator:
LCM(10π/5, 2π/5)
LCM(10π - 2π)/5
LCM(8π/5) = 8π/5

Therefore, the period of g(x) is 8π/5.

Step 2: Determining the Domain and Range
The domain of a function is the set of all possible input values (x) for which the function is defined. In this case, there are no restrictions on the domain of g(x) since both cosine and sine functions are defined for all real numbers.

So, the domain of g(x) is (-∞, +∞).

The range of g(x) is the set of all possible output values (y) that the function can produce. Since both cosine and sine functions have ranges between -1 and 1, the range of g(x) will also be bounded between certain values.

To determine the exact range of g(x), we can rewrite g(x) using trigonometric identities:

g(x) = 2cos(2x) + 3sin(5x)
= 2cos(2x) + 3(sin(2x)cos(3x) + cos(2x)sin(3x))
= 2cos(2x) + 3sin(2x)cos(3x) + 3cos(2x)sin(3x)
= (2 + 3cos(3x))sin(2x) + 3cos(2x)sin(3x)

From this form, we can see that the range of g(x) will depend on the values of cos(3x) and sin(3x). Since both cosine and sine functions oscillate between -1 and 1, the range of g(x) will be greater than or equal to the minimum value of (2 + 3cos(3x)) and less than or equal to the maximum value of (2 + 3cos(3x)).

To determine the minimum and maximum values of (2 + 3cos(3x)), we need to find the range of cos(3x), which is -1 to 1. Therefore,

Minimum value = 2 + 3 * (-1) = -1
Maximum value = 2 + 3 * (1) = 5

Therefore, the range of g(x) is -1 ≤ g(x) ≤ 5.

To summarize:
- The function g(x) = 2cos(2x) + 3sin(5x) is periodic, with a period of 8π/5.
- The domain of g(x) is (-∞, +∞).
- The range of g(x) is -1 ≤ g(x) ≤ 5.