Questions: 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. g(x)=x^6-2x Determine whether the function is even, odd, or neither. Choose the correct answer below. - even - odd - neither Determine whether the graph of the function is symmetric with respect to the y-axis, the origin, or neither. Select all that apply. - neither - y-axis - origin

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. g(x)=x^6-2x Determine whether the function is even, odd, or neither. Choose the correct answer below. - even - odd - neither Determine whether the graph of the function is symmetric with respect to the y-axis, the origin, or neither. Select all that apply. - neither - y-axis - origin
Transcript text: Algebra/Geometry Prerequisite Quiz lab.pearson.com/Student/PlayerTest.aspx?testld=2680834428centerwin=yes h 36 - Winter 2025 Daniel Agbelusi 01/08/25 7:40 PM This quiz: 12 Quiz: Algebra/Geometry Prerequisite Quiz Question 2 of 12 point(s) possible This question: 1 point(s) possible Submit quiz 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. \[ g(x)=x^{6}-2 x \] Determine whether the function is even, odd, or neither. Choose the correct answer below. even odd neither Determine whether the graph of the function is symmetric with respect to the $y$-axis, the origin, or neither. Select all that apply. neither $y$-axis origin Next cessibility: Investigate Notes Comments JAN
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Solution

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Solution Steps

Step 1: Determine if the Function is Even, Odd, or Neither

To determine if the function \( g(x) = x^6 - 2x \) is even, odd, or neither, we need to evaluate \( g(-x) \) and compare it to \( g(x) \).

  • A function is even if \( g(-x) = g(x) \).
  • A function is odd if \( g(-x) = -g(x) \).

Calculate \( g(-x) \):

\[ g(-x) = (-x)^6 - 2(-x) = x^6 + 2x \]

Now, compare \( g(-x) \) with \( g(x) \):

  • \( g(x) = x^6 - 2x \)
  • \( g(-x) = x^6 + 2x \)

Since \( g(-x) \neq g(x) \) and \( g(-x) \neq -g(x) \), the function is neither even nor odd.

Step 2: Determine Symmetry of the Graph

To determine the symmetry of the graph, we consider the following:

  • A graph is symmetric with respect to the \( y \)-axis if the function is even.
  • A graph is symmetric with respect to the origin if the function is odd.

Since the function is neither even nor odd, the graph is neither symmetric with respect to the \( y \)-axis nor the origin.

Final Answer

  • The function is \(\boxed{\text{neither}}\) even nor odd.
  • The graph is symmetric with respect to \(\boxed{\text{neither}}\) the \( y \)-axis nor the origin.
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