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Problem 78 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) = 1 + 3x^3 - x^5 $

Answer

Neither odd or even.

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Video Transcript

all right here we have function f of X, and we want to determine if it's odd, even or neither. So remember that for odd functions, opposite X values have opposite. Why values? So the grafts have origin, symmetry and for even functions opposite X values have the same y value. So the grafts have y axis symmetry. So let's figure out what we get for f of the opposite of X for this function. So we're going to need to plug the opposite of X in everywhere we see Annex and it will look like this. Now when we simplify that, we get one minus three x cubed plus X to the fifth power. Now that does not look like the original function, and it does not look like the opposite of the original function. So this is not an even function, and it's not an odd function. It's neither. Okay, so we should be able to verify that if we look at the graph, we shouldn't see origin symmetry, and we shouldn't see why. Access symmetry. So we type in the function. I have it there in my y one, and then I take a look at the graph. Now, if you just grants glance at this quickly, you might accidentally think that it has origin symmetry. But notice that it's shifted up one unit, so it's not actually rotating about the origin. It's rotating about the point at a height of one, if you were to actually look at the one that rotates about the origin without the plus one in it, you see that when the red one has origin symmetry, but not the one that we were focused on.

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