Question

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 + 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**.