Let's solve the given function [tex]\( y = \frac{2 \cos(3x)}{x^3} \)[/tex] step-by-step.
1. Understanding the structure of the function:
The function [tex]\( y \)[/tex] is given in the form of a fraction:
[tex]\[
y = \frac{2 \cos(3x)}{x^3}
\][/tex]
Here, the numerator is [tex]\( 2 \cos(3x) \)[/tex] and the denominator is [tex]\( x^3 \)[/tex].
2. Numerator Analysis:
The numerator of the function is [tex]\( 2 \cos(3x) \)[/tex], which involves a cosine function that is being scaled by 2 and has an argument of [tex]\( 3x \)[/tex]. This indicates a trigonometric manipulation.
3. Denominator Analysis:
The denominator of the function is [tex]\( x^3 \)[/tex], which indicates that as [tex]\( x \)[/tex] increases or decreases from zero, the denominator grows much more quickly than the numerator, affecting the behavior of [tex]\( y \)[/tex].
4. Combining both parts:
When we put both parts together, we get the fraction:
[tex]\[
y = \frac{2 \cos(3x)}{x^3}
\][/tex]
This function represents a ratio where the cosine function [tex]\( \cos(3x) \)[/tex] oscillates between -1 and 1, scaled by a factor of 2, and then this entire term is divided by [tex]\( x^3 \)[/tex].
Visualizing the function:
- As [tex]\( x \)[/tex] approaches zero, the value of [tex]\( y \)[/tex] will depend on the behavior of both the numerator and the denominator near zero.
- As [tex]\( x \)[/tex] becomes very large or very small (positively or negatively), the [tex]\( x^3 \)[/tex] term in the denominator will dominate, causing [tex]\( y \)[/tex] to approach zero.
Special cases to consider:
- For [tex]\( x = 0 \)[/tex], the function is undefined because division by zero is undefined.
- For [tex]\( x \)[/tex] values where [tex]\( x \neq 0 \)[/tex], the function will produce a valid numerical result based on the cosine value and the cube of [tex]\( x \)[/tex].
In summary, the function [tex]\( y \)[/tex] is expressed succinctly as:
[tex]\[
y = \frac{2 \cos(3x)}{x^3}
\][/tex]
which provides a clear fraction where the trigonometric cosine function, scaled, interacts with a cubic polynomial in the denominator. The behavior of [tex]\( y \)[/tex] is critically dependent on the interplay between the rapidly oscillating numerator and the cubic growth in the denominator.
