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

Question

Determine whether each function is even, odd, or neither. See Example 5.

ƒ(x) = -x³ + 2x

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 negative five X rays to the fifth plus 17 X. Our answer choices are answer choice. A, an odd function. Answer choice B an even function and answer choice C neither. So what are odd and even functions we recall from previous lessons that an odd function is when we would input F of negative X, it would yield a negative F of X. We recall that an even function would be that if we put in for F of negative X again, we will get F of X and for neither one, if we put in that negative X, so we have F of negative X that will not equal negative F of X and F of negative X will not equal F of X. So that's great. But what does that mean? Well, for all of them, we're just going to put in a negative X anywhere we see an X and then we see what happens if all of the signs change, then that's an odd function. All of the terms signs change. If none of the terms signs change after we simplify it, if it goes right back to where it started after we simplify, then that's an even function. Now, for neither it means that some of the terms signs change and some of the terms signs do not change. So let's see what we've got here. We'll put in a negative X for X, we have F of negative X equals negative five multiplied not by X to the fifth, but now by negative X to the fifth plus 17 multiplied not by X but by negative X. Now we simplify with this negative X raised to the fifth power. It's being raised to an odd power and when that happens, it keeps its negative. So this is a negative five multiplied by negative X to the fifth. And then we still have this plus 17 multiplied by negative X will simplify with multiplication on this line. So now we have a negative five multiplied by a negative that's going to become a positive five X rays to the fifth. And here we have a positive 17 multiplied by a negative X and that's going to now become a negative 17 X. So what have we got? Did all of the signs change none of the signs, some of the signs? Well, we started off with a negative five X to the fifth. Now we have a positive five X to the fifth. We started off with a positive 17 X and then we have a negative 17 X. All of the signs changed. This is an odd function. We look at our answer choices and this 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. ƒ(x) = -x³ + 2x

Key Concepts

Even and Odd Functions

Polynomial Functions

Testing for Evenness or Oddness

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