A function **cannot** be both even and odd at the same time unless it is the **zero function**.

### Here's why:

- **Even function**: A function \( f(x) \) is even if:
  \[
  f(x) = f(-x) \quad \text{for all} \, x
  \]
  In other words, the graph of the function is symmetric about the y-axis.

- **Odd function**: A function \( f(x) \) is odd if:
  \[
  f(x) = -f(-x) \quad \text{for all} \, x
  \]
  This means the graph of the function has rotational symmetry about the origin (it looks the same if rotated 180° around the origin).

### Can a function be both even and odd?

For a function to be both even and odd, it must satisfy both conditions simultaneously. Let's analyze this:

1. From the **even** condition: \( f(x) = f(-x) \).
2. From the **odd** condition: \( f(x) = -f(-x) \).

Setting \( f(x) = f(-x) \) equal to \( f(x) = -f(-x) \) gives:
\[
f(x) = -f(x)
\]

This equation implies that \( 2f(x) = 0 \), or \( f(x) = 0 \) for all \( x \).

Thus, the only function that is both even and odd is the **zero function**, \( f(x) = 0 \), because it satisfies both conditions trivially.

### Conclusion:
A function can only be both even and odd if it is identically zero.

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Would you like further clarification on even or odd functions, or any other related concepts? 

### Related Questions:
1. What are examples of even functions and odd functions?
2. How do transformations affect the symmetry of a function?
3. What is the significance of symmetry in real-world applications of functions?
4. Can a piecewise function be both even and odd?
5. How do you prove if a function is even or odd?

**Tip**: When working with functions, remember that symmetry properties can often simplify solving and graphing them, especially when they have specific geometric interpretations.

Explore whether a function can be both even and odd, with a focus on symmetry properties. Learn the conditions for even and odd functions, and discover why only the zero function satisfies both criteria. Suitable for students in Grades 9-12.

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