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
g(x)=x + 3x
Determine whether the function is even, odd, or neither. Choose the correct answer below.
even
neither
odd
Answers
GPT-4o mini
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**.
\[
g(x) = x + 3x = 4x
\]
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**.
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