In Exercises 9–16, determine whether the function is even, odd...

Question

In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = x - sin x

Accepted Answer

Hello. In this video, we are going to be determining whether the given function is even, odd, or neither. The function given to us is Y is equal to X plus cosine of X. Now, before we approach this problem, let's go ahead and rewrite our function in terms of F of X. We will go ahead and replace Y with F of X, and the function will be rewritten as F of X is equal to X plus cosine of X. Now, in order to approach this problem, let's just go ahead and review the conditions for even or odd functions. If a function is odd, then what we need to do is we need to plug in the quantity negative X into our function. If the output comes out to be a negative version of our original function, then the function itself is odd. For even functions, if we take the same input, negative X, and plug it into our function, if the output is the original function, then the function is even. So, we will go ahead and start this problem by plugging in negative X into our function. That is going to give us F of negative X is equal to negative X plus cosine of negative X. Now recall that cosine isn't even function. There is an identity that states that if we have cosine of negative X, because it's an even function, we can rewrite it as cosine of X. So we can simplify the second term to be cosine of X, allowing our function to be rewritten as F of negative X is equal to negative X plus cosine of X. Now, notice that when we plugged in the version, or plugged in the value of negative X, the version of our function does not match the original function. So we can conclude that the function itself is not going to be even. Now we want to do is you want to go ahead and check to see if the function is odd. We can start by factoring out a -1 from the F of negative X function. If we factor out a -1, we get the function negative X minus cosine of X. But even after factoring out the negative one, we don't have the negative version of our original function. So what this means is that the function is not odd as well. And since we determined that the function is neither even or odd, then the solution to this problem is that the function itself is neither even or odd, and that is going to be the answer to this problem. So I hope this video helps you in understanding how to determine even your odd functions, and I will go ahead and see you all in the next video.

In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = x - sin x

Key Concepts

Even Functions

Odd Functions

Neither Even Nor Odd Functions

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