Even and Odd Functions

In Ex...

Question

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

g(x) = x⁴ + 3x² − 1

Accepted Answer

Welcome back, everyone. Determine if the function h of X equals 2x the power of 6 minus 4 x cubed plus 2 is even odd or neither. We're given 4 answer choices A says odd, B even, C neither, and D cannot be determined. If we identify if a specific function is even, odd, or neither, we have to begin with evaluating H of minus X. And then we have to consider two cases. If this returns as the original function H of X, then the function is even. If it returns as negative of each of X, then the function is odd. Otherwise, the function is neither. So let's go ahead, so evaluate each. Of negative X. This means that for every X that we have in the age of X, we are replacing it with negative X. So we get 2 multiplied by negative X. Race the power of 6 minus 4 multiplied by negative X. Raise the power of 3 + 2. Now let's simplify it. We get 2 X the power of sex because we are raising a negative term to an even exponent. Minus 4 multiplied by. Negative X cubed. In this case, we're raising a negative term to an odd exponent so we can factor out the negative sign, which returns us. Plus 4. XQ And then we're adding 2. So overall, we're getting 2 X to the power of 6 + 4 XQ plus 2. Now, is this the original function? We have 2 X, the power of 6, the same term, plus 4 X cubed. This is not the same term, right? It has an opposite sign, so it is not equal to each of X, meaning it is not even. And now we also want to factor out the negative sign to check if it's odd, or simply speaking, If this function is odd, then in our result, every term must have the opposite sign. Let's notice that 2 X is the power of 6 has the same the same sign, then we are flipping the sign of -4 X cubed into a positive sign, and plus 2 remains the same. So we have only flipped the sign of the second term and not each sign, right? Meaning it is not equal. To negative H of X, meaning the function is not odd. Therefore, it is neither right because it's not even and not odd, and our final answer corresponds to the answer choice C. It is neither. Thank you for watching.

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

g(x) = x⁴ + 3x² − 1

Key Concepts

Even Functions

Odd Functions

Neither Even Nor Odd Functions

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