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 = 1 - cos x

Accepted Answer

Welcome back everyone. For the given function is it even odd, or neither Y equals 2 minus sine of X. Let's recall the S of symmetry. What we have to do is define our original function F of X, then we have to identify F of negative X and then check two conditions. If it is equal to. F of X, then we can conclude that the function is even. If F of negative X is going to become negative F of X, then we have an odd function. If none of these conditions holds, then we can simply tell that the function is neither even nor odd. So we're going to use this test of symmetry. Let's evaluate f of negative X. So we're taking 2 minus sign of negative X. Simply speaking, We are substituting negative X for every X that we have, right? And now let's simplify what we have to do for this problem is simply understand that sign is an odd function. This comes from trigonometry, and that's because sign is symmetrical or relative to the origin, right? Essentially if we plot sign, we will notice that. The sine function is symmetrical relative to the origin, so it is an odd function. And because it's an odd function, it means that sign of negative X. Is equal to negative sign of x. Based on the definition of an odd function. So we're going to use this definition. And this gives us 2 less sign of Xs. That's because we get negative, negative sign off X. So now is it even? Well, is it equal to? 2 minus sin of X. It is not equal to the original function because we have that negative sign, right? So we're going to say that it is not equal to. The original Function F of X. Which means that It is not even. And now is it equal to? Negative of the original function. If we evaluate negative of the original function, negative F of X, we're going to get -2 plus sign of X. In other words, we're just flipping the sign of each term, right? So is it Equal to negative of X. Well, we still have sin of x with a positive sign, but we have 2 and 2, which essentially do not correspond to the same value, meaning it is not equal to negative F of X as well. And if it's not equal to negative F of X, it's not odd, right? So it's not even not odd meaning it's neither. And that's our final answer to this problem. Thank you for watching.

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

𝔂 = 1 - cos x

Key Concepts

Even Functions

Odd Functions

Neither Even Nor Odd Functions

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