Let's analyze the functions one by one to determine which are one-to-one (injective). A function is one-to-one if different inputs always map to different outputs.

• A. $f: \mathbb{R} \to \mathbb{R}$, defined by $f(x) = x^3 - x$: This function is not one-to-one. The function $f(x) = x^3 - x$ has critical points at $x = -1, 0, 1$, where the derivative $f'(x) = 3x^2 - 1$ equals zero. Also, $f(x)$ maps multiple values of $x$ to the same output, such as $f(1) = 0$ and $f(-1) = 0$, so it fails the horizontal line test.

• B. $f: \mathbb{R}^2 \to \mathbb{R}^2$, defined by $f(x, y) = (x + y, 2x + 2y)$: This function is not one-to-one. Both components of the output are linear combinations of $x + y$. This means that if $x + y$ is the same for different pairs $(x, y)$, the outputs will be the same. For example, $f(1, 0) = f(0, 1) = (1, 2)$, so it’s not injective.

• C. $f: \mathbb{R}^2 \to \mathbb{R}^2$, defined by $f(x, y) = (x + y, x - y)$: This function is one-to-one. To see this, consider two distinct inputs $(x_1, y_1)$ and $(x_2, y_2)$. If $f(x_1, y_1) = f(x_2, y_2)$, then: $x_1 + y_1 = x_2 + y_2 \quad \text{and} \quad x_1 - y_1 = x_2 - y_2.$ Solving these two equations, we find that $x_1 = x_2$ and $y_1 = y_2$, so the function is injective.

• D. $f: \mathbb{R}^3 \to \mathbb{R}^3$, defined by $f(x, y, z) = (x - y, y - z, x - z)$: This function is one-to-one. Similarly, if $f(x_1, y_1, z_1) = f(x_2, y_2, z_2)$, we would get: $x_1 - y_1 = x_2 - y_2, \quad y_1 - z_1 = y_2 - z_2, \quad x_1 - z_1 = x_2 - z_2.$ Solving these linear equations gives $x_1 = x_2$, $y_1 = y_2$, and $z_1 = z_2$, so the function is injective.

• E. $f: \mathbb{R} \to \mathbb{R}$, defined by $f(x) = x^3$: This function is one-to-one. The derivative $f'(x) = 3x^2$ is always positive (except at $x = 0$, where it is zero), meaning the function is strictly increasing, and thus injective.

Final Answer:

• C. and D. are correct choices, which are indeed one-to-one functions.

Would you like further clarifications or details? Here are some follow-up questions:

1. Can you explain why the derivative helps in determining if a function is one-to-one?
2. What does it mean for a function to pass the horizontal line test?
3. How would you check if a function from $\mathbb{R}^n \to \mathbb{R}^m$ is one-to-one in general?
4. Why are the linear combinations in B problematic for injectivity?
5. Can a non-linear function be one-to-one?

Tip: For polynomial functions like those in options A and E, checking the derivative is a quick way to see if the function is strictly increasing or decreasing, which implies it is one-to-one.