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.

Question

​f(x)equalsx cubed plus 6 x
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Part 1
Determine whether the function is​ even, odd, or neither. Choose the correct answer below.
even
neither
odd
Part 2
Determine whether the graph of the function is symmetric with respect to the​ y-axis, the​ origin, or neither. Select all that apply.
origin
​y-axis
neither

Answer 1

To determine whether the function \( f(x) = x^3 + 6x \) is even, odd, or neither, we will analyze \( f(-x) \):

1. Calculate \( f(-x) \): \[ f(-x) = (-x)^3 + 6(-x) = -x^3 - 6x \]

2. Compare \( f(-x) \) and \( f(x) \):

   • An even function satisfies \( f(-x) = f(x) \).
   • An odd function satisfies \( f(-x) = -f(x) \).

   We found: \[ f(-x) = -x^3 - 6x = -(x^3 + 6x) = -f(x) \] Since \( f(-x) = -f(x) \), the function \( f(x) \) is odd.

Part 1

The function is: odd

Part 2

To determine the symmetry of the graph:

• A function is symmetric about the y-axis if it is even.
• A function is symmetric about the origin if it is odd.

Since we have established that \( f(x) \) is odd, the graph of the function is symmetric with respect to the origin.

Final Answers

• Part 1: The function is odd.
• Part 2: The graph is symmetric with respect to the origin.