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 = sec x tan x

Accepted Answer

Hello. In this video, we are going to be determining whether the given function is even or odd, or neither. So, the function given to us in this problem is Y is equal to sin of X multiplied by cosine of X. In order to start this problem, let's first replace the Y symbol with F of X. We can rewrite the equation to be F of X is equal to sin of X multiplied by cosine 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. We will go ahead and start with the test for oddness. Now, in order for a function to be odd, if you have a function F of X and you plug in the value of negative X, if you simplify the function and it has the form negative F of X, then it is classified as an odd function. Now the test for evenness. Is a similar argument, but separate. The test for even states that if you plug in the value of negative X and you get out the original function, then your function is going to be even. So, in order to start this problem, let's go ahead and plug in negative X into our given function. So we will have F of negative X equal to sin of negative X multiplied by cosine of negative X. Now, let's quickly recall that sign is an odd function. Since sign is an odd function, we have the identity that sign of negative X can be rewritten as negative sign of X. And for cosine, cosine isn't even function. The identity cosine of negative X can be rewritten as cosine of X because it is an even function. So, what we will go ahead and do is apply these identities, and we can rewrite F of negative X to be negative sign of X multiplied by cosine of X. So, well, there's one thing to notice about the output of this function. First, we have sin of x multiplied by cosine of X, which matches the form of the original F of X. But notice that the difference between this output and our original function is that it has a negative sign in front of it. What this means is that the output matches negative F of X, and that satisfies the test for oddness of the given function. So what this means is that the function that is given to us is an odd function. And that is going to be the solution to this problem. So I hope this video helps you in determining if 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 = sec x tan x

Key Concepts

Even and Odd Functions

Trigonometric Functions

Function Composition and Transformation

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