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

Question

Determine whether each function is even, odd, or neither. See Example 5. ƒ(x) = x³ - x + 9

Accepted Answer

Welcome back. I am so glad you're here. We're asked for the function below to determine if it is even odd or neither. Our function is F of X equals X raised to the fifth power minus three X plus 11. Our answer choices are answer choice. A, an odd function, answer choice B and even function and answer choice. C neither. All right. So what are even odd and neither functions we recall from previous lessons that an odd function will exist when we take F of negative X and it yields negative F of X. An even function will exist when we take F of negative X and it yields F of X and neither exists when neither of those situations exist when we take F of negative acts. And that does not equal negative F of X. And when we take F of A or F of negative X and it does not equal F of X for neither some signs change and some do not. All right. So this is the technical definition. But what does all of this mean? Well, it means that we're going to plug in a negative X or X and see what we get. So instead of F of X will have F of negative X that equals instead of X ray to the fifth power that will be a negative X raise to the fifth power then minus three multiplied not by X but by negative X still plus 11. Now we simplify and we see which signs change if any. So negative X rays to the fifth power that's raised to an odd number for its exponent. So that one is going to change that one's going to simplify it to be a negative x-rays to the fifth power. Now multiplying negative three by negative X that's going to become a positive three X, then we bring down our plus 11. Now we notice that some of our signs change this negative X to the fifth instead of a positive X to the fifth. And this positive three X instead of this negative three X, but one term sign did not change, we still have a positive 11. So we have neither the exact opposite of what we started with an odd function. It's where every sign changes for each term. So we don't have that and we don't have exactly what we started with. Either some of the signs did change and even function where would be where no signs change. What we do have is a neither function. This is where some signs change and some do not. We have neither even nor odd. We'll look at our answer choices and this matches with answer choice. See, well done. We'll catch you on the next one.

Determine whether each function is even, odd, or neither. See Example 5. ƒ(x) = x³ - x + 9

Key Concepts

Even and Odd Functions

Function Evaluation and Substitution

Polynomial Function Properties

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