This given problem is dealing with symmetry and even functions.
A graph of a function is symmetric about the y-axis if and only if the function is an even function. An even function is one where for all x in the domain f(-x) = f(x).
We'll apply this condition to our given function 3y-1/[(yⁿ)(3y+1)]. In this case, y is our variable instead of x, but the same principle applies.
Let's try to substitute variable y = -y and compare this expression with the original function:
Our function after substituting -y for y becomes:
f(-y) = 3(-y)-1/(-(yⁿ)(3(-y)+1)]
This does not simplify to our original function:
f(y) = 3y-1/[(yⁿ)(3y+1)]
Therefore, comparing these two expressions, it's clear that they aren't equal:
f(-y) ≠ f(y)
This leads us to conclude that function 3y-1/[(yⁿ)(3y+1)] is not symmetric about the y-axis. In other words, it's not an even function.
Remember, the question asked us to find the value of f(n) giving that the function is symmetric about the y-axis. But, since we've demonstrated that it's not symmetric, the task of finding f(n) based on symmetry is impossible. In this case, the issue lies not in our calculation, but rather in the incorrect initial statement of the problem.
Hence, the answer is that the problem task statement is incorrect; the function is not symmetric about the y-axis.