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 cos x

Accepted Answer

Hello. In this video, we are going to be determining whether the given function is even odd or neither. So the function given to us is Y is equal to X multiplied by sin of X. Before we start this problem, let's go ahead and rewrite the function in terms of F of X. We will replace Y on the left-hand side with the symbol F of X, and that will allow us to rewrite the function, as F of X is equal to X multiplied by sin of X. Now, in order to approach this problem, let's just quickly review what it means for a function to be even or odd. So for an odd function, if we were to take the value of negative X and plug it into our function, if the output is a negative version of our original function, then that determines that the function is going to be odd. Similarly, for the even functions, if we take the value of negative X and plug it into our function, if the output is the original function, then the function itself is going to be even. So we will approach this problem by plugging in the value of negative X into our function. That is going to allow us to rewrite the function, as F of negative X is equal to negative X multiplied by sign of negative X. Now recall the identity for sign of negative X. Since sign is an odd function, sign of negative X is equivalent to negative sign of X. So what we will go ahead and do is use this identity to rewrite sign of negative X. That will allow us to rewrite the F of negative X function to be negative X multiplied by negatives of X. And since we have two coefficients of -1, multiplying them together will give us a coefficient of 1, allowing us to rewrite the function as X multiplied by sin of X. Notice that when we plug in negative X and we simplify the function, the output is the same function as the original function given to us. What this means is that the conditions for the even function is satisfied, and the solution to this problem is that the function is going to be even. So I hope this video helps you in determining how a function is even or odd, 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 cos x

Key Concepts

Even Functions

Odd Functions

Function Analysis

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