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

g(x)=x + 3x

Determine whether the function is even, odd, or neither. Choose the correct answer below.

even

neither

odd

Answer 1

To determine whether the function \( g(x) = x + 3x \) is even, odd, or neither, we first simplify the function:

\[ g(x) = x + 3x = 4x \]

Now we will evaluate \( g(-x) \):

\[ g(-x) = 4(-x) = -4x \]

Next, we compare \( g(-x) \) with \( g(x) \) and \(-g(x)\):

1. Check if \( g(x) \) is even:

   • A function is considered even if \( g(-x) = g(x) \).
   • Here, \( g(-x) = -4x \) and \( g(x) = 4x \).
   • Since \( g(-x) \neq g(x) \), the function is not even.

2. Check if \( g(x) \) is odd:

   • A function is considered odd if \( g(-x) = -g(x) \).
   • Calculate \(-g(x)\):
   • \( -g(x) = -4x \).
   • We find that \( g(-x) = -4x\) which is equal to \(-g(x)\).
   • Since \( g(-x) = -g(x) \), the function is odd.

Since the function is odd, it is also symmetric with respect to the origin.

Final answers:

• The function is odd.
• The function's graph is symmetric with respect to the origin.