Determine whether each function is even, odd, or neither. See ...

Question

Determine whether each function is even, odd, or neither. See Example 5. 1 ƒ(x) = x + —— x⁵

Accepted Answer

Welcome back. I am so glad you're here. We're told for the function given below determine if it is even odd or neither. Our function is F of X equals three X plus four divided by X rays to the seventh power. Our answer choices are answer choice. A, an odd function, answer choice B and even function and answer choice. C neither. So what are odd and even functions will we recall from previous lessons that an odd function is found when we take F of negative X and it yields negative F of X and an even function is when we take F of negative X and it yields F of X. Neither is when some of the signs change and some don't. So what does that mean? All right. So it means that we're going to be plugging in to find out if it's odd or even, we're gonna be plugging in a negative X for our function. So anywhere there was an X before we're now gonna put in a negative X, then we simplify and see what we get. So we'll have F of negative X which will equal three multiplied no longer by X. But by negative X plus four divided by not X raised to the seventh power, but negative X raised to the seventh power. Now we simplify, there's not a whole lot to simplify. But for this three multiplied by negative X that becomes a negative three X. And for this denominator, this negative X raised to the seventh power. Well, when that's raised to an odd number, it is going to end up changing its sign to being negative. So this would be plus four divided by now negative X race to the seventh power. And when we have that negative in the denominator like that, we could put that up in the numerator or we could put it right out in front, changing that sign from plus to minus. So we can write this as negative three X minus four divided by X raised to the seventh power. And we noticed that both of these signs are now negative. We could factor in negative out in front. So that would be a negative open parentheses, three X plus four divided by X rays to the seventh power. And look at that, that is just like what we started with except the whole thing is negative. And that is the very definition of an odd function when we get what we started with. But the whole thing is negative or in other words, the sign of every term changed. There were two terms here, the signs of both of them changed. If one of the signs had changed with the other, not that would have been a neither situation. And if none of the signs had changed, if they had reverted to what we'd started with, then we would have had an even situation. But because the whole thing changed, it is odd and that matches with answer choice. A well done. We'll catch you on the next one.

Determine whether each function is even, odd, or neither. See Example 5. 1 ƒ(x) = x + —— x⁵

Key Concepts

Even and Odd Functions

Function Substitution and Simplification

Properties of Polynomial and Rational Functions

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