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.
f(x) = x^5 + x
Determine whether the function is even, odd, or neither. Choose the correct answer below.
neither
even
odd
Determine whether the graph of the function is symmetric with respect to the y-axis, the origin, or neither. Select all that apply.
y-axis
neither
origin
Graph shape/text: Function f(x) = x^5 + x; the graph is an odd polynomial curve symmetric about the origin. position_hint: top-left
1. **State the problem:** We need to determine if the function $$f(x) = x^5 + x$$ is even, odd, or neither, and then identify the symmetry of its graph.
2. **Recall definitions:**
- A function is **even** if $$f(-x) = f(x)$$ for all $$x$$.
- A function is **odd** if $$f(-x) = -f(x)$$ for all $$x$$.
- If neither condition holds, the function is neither even nor odd.
3. **Calculate $$f(-x)$$:**
$$f(-x) = (-x)^5 + (-x) = -x^5 - x$$
4. **Compare $$f(-x)$$ with $$f(x)$$ and $$-f(x)$$:**
- $$f(x) = x^5 + x$$
- $$-f(x) = -(x^5 + x) = -x^5 - x$$
Since $$f(-x) = -f(x)$$, the function is **odd**.
5. **Determine symmetry:**
- Odd functions have graphs symmetric about the **origin**.
- Even functions have graphs symmetric about the **y-axis**.
Therefore, the graph of $$f(x) = x^5 + x$$ is symmetric about the **origin**.
**Final answers:**
- The function is **odd**.
- The graph is symmetric about the **origin**.