Even and Odd Functions

In Ex...

Question

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

f(x) = x² + x

Accepted Answer

Welcome back everyone. Determine if the given function is even, odd, or neither. G of X equals x the power of 31 minus X. We're given 4 answer choices as as odd, B even C neither, and D cannot be determined. If we want to identify the symmetry of a function, we have to evaluate G of negative X. And then we have to consider two cases. If this gives us the original function G of X, then we can conclude that the original function is even. If we end up with negative G of X, then the function is odd. Otherwise, the function doesn't contain a symmetry, right, and it is neither. So in this case, let's value G of negative X, which means that for every X that we have, we're going to include negative X. Which gives us negative X, raise the power of 31 minus negative X. Let's consider each term, starting with the first term, because we are raising a negative term to an odd exponent, we're going to get a negative term, right? So we can factor out the negative sign, and then we can raise X to the power of 31. And then we are subtracting a negative term and we can replace it with additions, so we're adding X. Now, what we can see is that this is not the original function simply by observing the change in each sign, right? Now let's go ahead and factor out the negative sign to check if it's possibly minus G of X. So if we factor a negative sign, this gives us minus X to the power of 31 minus X, and as we can see, the inner part of the expression corresponds to the original function G of X, so we can say that this is equal to minus G of X. And therefore, the given function is an odd function. We can conclude that the correct answer to this problem corresponds to the answer choice A. Thank you for watching.

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

f(x) = x² + x

Key Concepts

Even Functions

Odd Functions

Neither Even Nor Odd Functions

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