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Determine whether the function is even, odd, or neither. Then determine whether the function's graph is symmetric with respect to the y-axis, the origin, or neither. h(x) = x* - x6 Determine whether the function is even, odd, or neither. Choose the correct answer below. odd neither even

Determine whether the function is even, odd, or neither. Then determine whether the function's graph is symmetric with respect to the y-axis, the origin, or neither. h(x) = x* - x6 Determine whether the function is even, odd, or neither. Choose the correct answer below. odd neither even

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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--- ## Even, Odd, or Neither Functions ### Problem Statement: Determine whether the function is even, odd, or neither. Then determine whether the function's graph is symmetric with respect to the y-axis, the origin, or neither. Given function: \[ h(x) = x^4 - x^6 \] **Determine whether the function is even, odd, or neither. Choose the correct answer below:** - \(\circ\) odd - \(\circ\) neither - \(\circ\) even (Note: There are no graphs or diagrams present in the image related to this content.) --- ### Explanation: To determine whether a function is even, odd, or neither, you examine the function's behavior when \(x\) is replaced by \(-x\): - **Even Function:** A function \( h(x) \) is even if \( h(-x) = h(x) \). An even function is symmetric with respect to the y-axis. - **Odd Function:** A function \( h(x) \) is odd if \( h(-x) = -h(x) \). An odd function is symmetric with respect to the origin. - **Neither:** If the function does not satisfy either condition, it is neither even nor odd. **Steps to determine the type of function for \( h(x) = x^4 - x^6 \):** 1. Compute \( h(-x) \): \[ h(-x) = (-x)^4 - (-x)^6 \] 2. Simplify each term: \[ (-x)^4 = x^4 \quad \text{(because raising -x to an even power gives a positive result)} \] \[ (-x)^6 = x^6 \] 3. Combining these, we get: \[ h(-x) = x^4 - x^6 \] 4. Compare \( h(-x) \) and \( h(x) \): \[ h(-x) = x^4 - x^6 \] \[ h(x) = x^4 - x^6 \] Since \( h(-x) = h(x) \), the function \( h(x) \) is even, and thus symmetric with respect to the y-axis. Choose the answer "even" from the options provided.
Transcribed Image Text:--- ## Even, Odd, or Neither Functions ### Problem Statement: Determine whether the function is even, odd, or neither. Then determine whether the function's graph is symmetric with respect to the y-axis, the origin, or neither. Given function: \[ h(x) = x^4 - x^6 \] **Determine whether the function is even, odd, or neither. Choose the correct answer below:** - \(\circ\) odd - \(\circ\) neither - \(\circ\) even (Note: There are no graphs or diagrams present in the image related to this content.) --- ### Explanation: To determine whether a function is even, odd, or neither, you examine the function's behavior when \(x\) is replaced by \(-x\): - **Even Function:** A function \( h(x) \) is even if \( h(-x) = h(x) \). An even function is symmetric with respect to the y-axis. - **Odd Function:** A function \( h(x) \) is odd if \( h(-x) = -h(x) \). An odd function is symmetric with respect to the origin. - **Neither:** If the function does not satisfy either condition, it is neither even nor odd. **Steps to determine the type of function for \( h(x) = x^4 - x^6 \):** 1. Compute \( h(-x) \): \[ h(-x) = (-x)^4 - (-x)^6 \] 2. Simplify each term: \[ (-x)^4 = x^4 \quad \text{(because raising -x to an even power gives a positive result)} \] \[ (-x)^6 = x^6 \] 3. Combining these, we get: \[ h(-x) = x^4 - x^6 \] 4. Compare \( h(-x) \) and \( h(x) \): \[ h(-x) = x^4 - x^6 \] \[ h(x) = x^4 - x^6 \] Since \( h(-x) = h(x) \), the function \( h(x) \) is even, and thus symmetric with respect to the y-axis. Choose the answer "even" from the options provided.
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