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Problem 77 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^2 - x^4 $

Answer

Even function.

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Video Transcript

So here we have a 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 graphs will have origin, symmetry and for even functions. Opposite X values have the same y value, so the graphs will have y axis symmetry. So we went to find out what f of the opposite of exes and determine if it's the same as the original, the opposite or neither. So we substitute the opposite of X in for acts in our function, and we get one plus three times the opposite of X squared, minus the opposite of X to the fourth power. Now, when you're squaring or raising to the fourth, whether it's positive or negative, it's going to end up to be the same. So the opposite of X quantity squared is equivalent to X squared, and the opposite of X quantity to the fourth is equivalent to X to the fourth. So notice that what we have now looks exactly the same as what we started with. So that means that f of the opposite of X is equal to f of X, and that means that we haven't even function. So if we graph this, we should be seeing why. Access symmetry so we can grab a calculator. Type this in and graph, and there's a graph that does appear to have y axis symmetry.

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