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Numerade Educator

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Problem 74 Hard Difficulty

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^2}{x^4 + 1} $

Answer

Even function

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Video Transcript

here we have a function, and we want to determine 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 y value. And so they have y axis symmetry. So what we want to do with this function is determined what we get when we plug in the opposite X value so f of the opposite of X. We're going to substitute the opposite of X in for acts in the function, and then we'll simplify. And the opposite of X squared is equivalent to X squared, and the opposite of X to the fourth is equivalent to X to the fourth. So are simplified version of EFT of negative X is equivalent to F of X that tells us that our function is even now we can go ahead and graph it on a graphing calculator and verify that by looking for the y axis symmetry. So we type it into our calculators y equals menu, and then we graph, and that does appear to have symmetry across the Y axis, so that confirms that it is even