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.