Ask AI: Solve the following is either y-axis or origin symmetry or neither, X²/X³-X

First, let's rewrite the equation as f(x) = x²/(x³ - x). Let's test the function for y-axis symmetry. A function has y-axis symmetry (also known as 'even' function) if f(x) = f(-x) for all x in the function's domain. That is, the function's value is the same for the positive and negative values of x. Now, f(-x) = (-x)²/((-x)³ - (-x)). Simplifying further gives x²/(-x³ + x) = - x²/(x³ - x). So we find f(x) ≠ f(-x), so the function is not symmetric about the y-axis. Let's test the function for origin symmetry. A function has origin symmetry (also known as 'odd' function) if f(x) = -f(-x) for all x in the function's domain. This means that the function's value for -x is the negative of the function value at x. So let's test - f(-x): -f(-x) = -((-x)²/((-x)³ - (-x))) = -x²/(x³ - x) This also results in f(x) ≠ -f(-x), so the function is not symmetric about the origin either. In conclusion, the function f(x) = x²/(x³ - x) is neither symmetric about the y-axis nor the origin.