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 = x⁴ + 1

x³ - 2x

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 cubed plus 2 divided by X2 minus 3. Now, before we approach this problem, let's just quickly rewrite Y in terms of F of X. So we can rewrite the function to be F of X is equal to X cubed plus 2 divided by X2 minus 3. Now, let's just quickly review the the conditions for even or odd functions. If a function is odd. Then, if we plug in F of negative X, which means we take the value of negative X and plug it into our function, if the output comes out to be negative F of X or negative version of our original function, then the function itself is going to be odd. For even functions, if we plug in negative X into our original function, and the output is the original function, then the function itself is even. So, in order to start this problem, let's go ahead and plug in negative X into our F of X function. That is going to give us F of X is equal to negative X raise to the power of 3 + 2 divided by negative X raise to the power of 2 minus 3. Now what we need to do is we need to simplify the function. First, any quantity that is raised to a negative uh an odd exponent is going to be negative. So if we plug in X and raise it to the power of negative 3, Then that is going to give us negative X as the output. So the numerator is going to simplify to be negative X cubed plus 2. Now, whenever we square a negative quantity, such as negative X2d, an even powered exponent is going to turn that quantity positive. So, negative X squared is going to simplify to be positive X2. Then we will subtract 3 from the original denominator. So, the simplified form of F of negative X is going to be negative X cubed plus 2, divided by X2 minus 3. Now, if we compare this to the even function, notice that when we plug in negative X, we do not get out the original function. Since the form of the given of the output is different from the original function, then we can conclude that the function is not even. But what about the odd function? Well, we want our function to be -1, multiplied by original function. The only thing that we can do to the numerator is we can factor out a -1, but if we factor out a -1, we get negative X cubed minus 2 divided by X2 minus 3. But that is still a different version of the function that was originally given to us, so the function is neither odd. Since both of the criteria for even and odd functions are not satisfied, then the solution to this problem is that the function 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 on how to determine even or 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 = x⁴ + 1x³ - 2x

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

𝔂 = x⁴ + 1
x³ - 2x