The following picture is a graph of a function $f$. I am to determine whether the function is even, odd, or neither. I reasoned that $f$ is odd, because if the graph is rotated $\pi$ radians, the graph is reproduced perfectly. The graph is clearly not even, because it is not symmetrical with respect to the vertical axis. However, the solution states that $f$ is neither even or odd, and I do not understand why.
Best Answer
You're right that the graph has a symmetry - if you rotate about a certain point on the $y$-axis, you preserve the graph. Algebraically, it has the property: $$f(x)=2c-f(-x)$$ where $c=f(0)$ is the intersection of the graph and the $y$-axis. The condition of oddness is more strict, however, it demands: $$f(x)=-f(-x)$$ meaning the function must have this rotational symmetry about the origin, not about some arbitrary point. Note that oddness implies $f(0)=0$.