In Exercises 9–16, determine whether the function is even, odd...

Question

In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = x⁵ - x³ - x

Accepted Answer

Welcome back, everyone. In this problem, for the given function below, Y equals X 4 minus 2 X2 + 3, is it even, odd, or neither? A says the function is even, B the function is odd. C, the function is neither even nor odd, and the D says it cannot be determined. Now, let's first ask ourselves, what do we know about even and add functions? Well, for starters, we know that the function is even if F of negative X. Is equal to F of X. For all values of X in the domain, OK. So now, basically to know if our function is even, we can substitute negative X into the function into Y and then that will help us to see if it gives us or back our original function FFX. Now F of negative X, then if we let, or rather let's just make this clear here, if we let Y be equal to F of X. Then F of negative X. Is going to be equal to negative X. To the 4th Minus 2 multiplied by negative X squared. Plus 3. No negative X to the 4th E equals X 4th, minus 2 multiplied by negative X2 is gonna be X2. OK. So no, we get back X 4 minus 2 X2 + 3, which is equal to F of X. So what does this tell us? Therefore, OK, FFX is even. Because F of negative X is equal to F of X. Now let's figure out the next property. Is the function odd? What do we know about odd functions? A function is odd if F of negative X. Is equal to the negative value of F of X, or all the values of X. So again, now all we need to do is to find negative F of X and see if it equals F of negative X, for which we already have a value. Now what is negative FF X? Well, it just means that now we're going to make our entire function negative, so that's gonna be the negative value of X to the 4th minus 2 X2 + 3. And to make this negative, we can make each term, or we can imagine that we're multiplying each term by -1. So this is going to be negative X to the 4th plus 2 X2 minus 3. And no, remember F of negative X + X to the 4th minus 2X2 + 3. So that means negative F of X is not equal to F of negative X. Therefore, the function is not odd, OK? So, therefore, FF X is not odd. Thus, that means that the function is even. A is the correct answer. Thanks for watching everyone. I hope this video helped.

In Exercises 9–16, determine whether the function is even, odd, or neither.𝔂 = x⁵ - x³ - x

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In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = x⁵ - x³ - x