Question

Use the algebraic tests to check for symmetry with respect to both axes and the origin. y = x/(x² + 2). ​(A) The function is symmetric with respect to the x-axis but not the y-axis or the origin.

(B) The function is symmetric with respect to the y-axis but not the x-axis or the origin.
(C) The function is symmetric with respect to the origin but not the x-axis or the y-axis.
(D) The function is not symmetric with respect to the x-axis, y-axis, or the origin.

Answer 1

The function y = x/(x² + 2) is confirmed to be symmetric about the origin after performing algebraic tests, confirming it is an odd function. Thus, the correct answer is (C), being symmetric to the origin only.

Step-by-step explanation:

To check for symmetry concerning the axes and origin for the function y = x/(x² + 2), we use algebraic tests:

• For symmetry about the y-axis, we check if f(x) = f(-x).
• For symmetry about the x-axis, we check if -f(x) = f(x), which never holds for non-constant functions.
• For symmetry about the origin, we check if -f(x) = f(-x).

Let's perform these tests:

1. Symmetry about the y-axis: f(-x) = -x/((-x)² + 2) = -x/(x² + 2) = -f(x), so the function is not symmetric about the y-axis.

2. Symmetry about the x-axis would imply that the function is even, which is not possible for non-constant functions.

3. Symmetry about the origin: -f(x) = -[x/(x² + 2)] = -x/(x² + 2) = f(-x). This shows that the function is indeed symmetric about the origin, confirming it is an odd function.

The correct answer is (C) The function is symmetric concerning the origin but not the x-axis or the y-axis.