The question asks to explain why an even function, whose domain contains a nonzero number, cannot be a one-to-one function.

Explanation:

An even function is defined by the property: $f(x) = f(-x) \quad \text{for all } x \text{ in the domain}.$ This means that for every $x$, the function value at $x$ is the same as the function value at $-x$. As a result, different inputs $x$ and $-x$ produce the same output, violating the requirement for a function to be one-to-one (injective).

A function is one-to-one if: $f(x_1) = f(x_2) \quad \Rightarrow \quad x_1 = x_2.$ In the case of an even function, for any nonzero $x$, $f(x) = f(-x)$ but $x \neq -x$, so the function is not one-to-one.

Conclusion:

Since an even function produces the same output for two distinct inputs ($x$ and $-x$, where $x \neq 0$), it cannot be a one-to-one function.

Would you like further details or clarifications?

Here are 5 related questions to deepen your understanding:

1. What is the difference between an even and an odd function?
2. Can a function be both one-to-one and even? Why or why not?
3. How does the horizontal line test relate to determining if a function is one-to-one?
4. What are examples of common even functions and why are they not one-to-one?
5. How does restricting the domain of an even function impact its injectivity?

Tip: To test if a function is one-to-one, you can apply the horizontal line test. If any horizontal line intersects the graph more than once, the function is not one-to-one.