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) = 0.5x⁴ - 2x² + 6

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 3.7 X rays to the sixth power minus 13 X rays to the fourth power plus 29. Our answer choices are answer choice. A odd function. Answer choice B even function and answer choice C neither. All right. So what are odd and even functions? We'll have an odd function when we put in F of negative X and it yields negative F of X will have an even function when we put in F of negative X and it yields F of X. And we'll have neither one when we put in F of negative X and it does not equal negative F of X and we put in F of negative X and it does not equal F of X. So what does that mean? It means we're going to put in a negative X or our X value. So anywhere there's an X, we're gonna put in a negative X and then we see what happens if all of the signs change, it's going to be an odd function if no signs change. So that's none of the signs. For all of the terms we get. What we started with, that's going to be an even function. Neither would be where some of the terms signs change and some of the terms signs do not. So let's see what we've got. All right, we're putting an F of negative X that equals 3.7 multiplied by not X to the sixth, but now negative X raised to the sixth minus multiplied now by negative X raised to the fourth instead of X to the fourth and still plus 29. Now we simplify not much changes, but we've got this negative X race to the sixth power. Well, when we raise a negative to an exponent, that's even that negative gets canceled out. And so it's going to become positive. So we have 3.7 multiplied by a positive X to the sixth minus multiplied. And now we've got this negative X race to the fourth power, but again, it's raised to an even power. So those negatives get canceled out and this is still just going to be an X race to the fourth power and then bring down our plus 29. So did any of our signs change? Well, no, we're actually right back with what we started with. The first term is positive. The second term is negative and the third term is positive. So we have exactly what we started with. This is the very definition of an even function. We look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

Determine whether each function is even, odd, or neither. See Example 5. ƒ(x) = 0.5x⁴ - 2x² + 6

Key Concepts

Even and Odd Functions

Polynomial Function Symmetry

Substitution Method for Testing Parity

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