How does the symmetry of f(x) = p(x)/q(x) depend on the symmetry of p(x) and q(x

Question

How does the symmetry of f(x) = p(x)/q(x) depend on the symmetry of p(x) and q(x)? Select all true statements. There may be more than one correct answer. If p(x) is odd and q(x) is even, then f(x) is odd. If p(x) and q(x) are both even, then f(x) is even. If p(x) is odd and q(x) is even, then f(x) is even. If p(x) and q(x) are both odd, then f(x) is even. If p(x) or q(x) is neither even nor odd, then f(x) is neither even nor odd. If p(x) and q(x) are both odd, then f(x) is odd. If p(x) is even and q(x) is odd, then f(x) is even. If p(x) is even and q(x) is odd, then f(x) is odd.

Explanation / Answer

If p and q are odd:
f(-x) = p(-x)/q(-x) = [-p(x)]/[-q(x)] = p(x)/q(x) = f(x)

If p is even and q is odd:
f(-x) = p(-x)/q(-x) = p(x)/[-q(x)] = -p(x)/q(x) = -f(x)

If p is odd and q is even:
f(-x) = p(-x)/q(-x) = [-p(x)]/q(x) = -p(x)/q(x) = -f(x)

If p and q are even:
f(-x) = p(-x)/q(-x) = [p(x)]/[q(x)] = p(x)/q(x) = f(x)

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