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Determine whether $ f $ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

$ f(x) = \dfrac{x}{x^2 + 1} $

Odd function

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Okay, so we have f of X equals X over X squared plus one. And we want to know if it's odd, even or neither. So remember that for odd functions, opposite X values give you opposite. Why values? And they have origin, symmetry and for even functions opposite X values give you the same. Why values? And so they have y axis symmetry. So let's investigate for our function. And let's figure out what f of the opposite of exes will it be the same or the opposite, or neither compared to the original. So what we need to do is substitute the opposite of X in for X in our function, and it will look like that. And then we can simplify. And we know that the opposite of X squared is just equivalent to X squared. So this looks like the opposite of our original function. Therefore, since half of the opposite of X equals the opposite of F of X, we have an odd function. So now when we look at the graph on the calculator, we should expect to see something that has origin symmetry. So we grab a calculator and type in the function and I'm going to try Zoom six for a standard negative 10 to 10 window. And here's how it looks. We could even change the window dimensions if we want. Maybe different y values. Let's say negative 3 to 3. Okay, that definitely looks like it has symmetry about the origin, so that confirms that it is an odd function.