Given:
x belongs to R

To find:
Interval in which 3cos(4x-5) lies

Solution:
To determine the interval in which 3cos(4x-5) lies, we need to analyze the range of values that cosine can take.

Range of Cosine Function:
The cosine function, cos(x), has a range of values between -1 and 1. It oscillates between these two values as the input angle varies.

Analysis of 3cos(4x-5):
1. The function 3cos(4x-5) is a composition of the cosine function and a linear transformation: 4x-5.
2. The linear transformation does not affect the range of the cosine function; it only shifts and scales the graph.
3. Since the range of the cosine function is between -1 and 1, the range of 3cos(4x-5) will be between -3 and 3.
4. This means that no matter the value of x, 3cos(4x-5) will always lie between -3 and 3.

Interval:
Therefore, the interval in which 3cos(4x-5) lies is (-3, 3), where the parentheses indicate that the interval does not include the endpoints.

Summary:
- The function 3cos(4x-5) will always have values between -3 and 3, regardless of the value of x.
- Thus, the interval in which 3cos(4x-5) lies is (-3, 3).