Can a function be both even and odd? Give reasons for your ans...

Question

Can a function be both even and odd? Give reasons for your answer.

Accepted Answer

Welcome back, everyone. Choose the function that is neither odd nor even. We're given the four answer choices. A says FX equals X2 minus X to the power of 6. BFFX equals sin of x the power of 4. CFX equals tangent of X minus X cubed, and D F of X equals the absolute value of X minus X to the power of 7. So let's analyze each function. Let's begin with A. What we want to do is simply valid F of negative X. So we're going to get negative of X2 minus negative of X, raise the power of 6. And because we're raising two Xs that have a negative sign in front to even powers, we can simply write that this is X2 minus X the power of 6, right? Because essentially whenever we raise A negative number to an even exponent, we get a positive number so we can eliminate those negative signs and as we can see, this is the original function. It has the same expression as F of X, and the conclusion is that it is even right, because F of negative X returns F of X. Now B What we're going to do is again evaluate F of negative X, we're going to get sin of negative. Xs raise the power of 4. The same idea applies here. We got sin of X to the power of 4, right? Because we are raising negative X to an even power which returns a positive number and therefore we can conclude that again. This is equal to the original function F of X, and it is even. So far we can eliminate the answer choices A and B. Let's continue with C. Let's evaluate F of negative X, we're going to get tangent. Of negative Xs. Minus negative of X cubed. For this one, we can rewrite tangent as sign of negative X. Divided by cosine of negative Xs. Minus X cubed. And then we have to recall that sine is an odd function, so we can rewrite sign sign off negative axis and negative sign off X. And cosine is an even function, so cosine of negative X is cosine x. And now if we raise negative X to an odd power, this returns as a negative numbers, we can say that we are subtracting. A negative number, meaning we are adding it, right? So we're adding X cubed. And now going back to the tangent function, we get minus tangent of X. Plus a cubed. Notice that we have flipped each sign, right? We have negative tangent plus X cubed. So this is essentially negative F of X, meaning it is an odd function. So the answer is not C. Meaning this must be D, so let's check F of negative X. We're going to get the absolute value of negative X. Minus Negative Xs raised the power of 7. So for this one, we're taking the absolute value of negative X, which is the same as the absolute value of positive X, right? Let's recall that the absolute value returns a positive number if it's negative. For example, the absolute value of -6 is equal to 6, which is basically the absolute value of 6 as well, right? So we can say that the absolute value of negative X is the absolute value of X, and in this case, We are subtracting negative X rays to an odd power which returns a negative number because it's an odd power, so subtracting a negative number gives us a positive number. So we're adding x to the power of 7. In this case, as we can see, we're not changing the sign of the absolute value of X, but we're flipping the sign of X to the power of 7. Now instead of being negative, it becomes positive. So this is not equal to F of X, it's not the same, and it's not equal to negative F of X, meaning it's neither, right? Therefore, we have concluded that the correct answer to this problem corresponds to the answer to the. The function that is neither odd nor even this function T because it it's F of negative X form doesn't return as F of X or negative F of X. Thank you for watching.

Can a function be both even and odd? Give reasons for your answer.

Key Concepts

Even Functions

Odd Functions

Mutual Exclusivity of Even and Odd Functions

Watch next