1. **State the problem:** Determine which of the given functions is one-to-one. 2. **Recall the definition:** A function is one-to-one (injective) if each output corresponds to exactly one input. Graphically, it passes the Horizontal Line Test. 3. **Analyze each function:** - A. $f(x) = 3x^3 + 1$ - This is a cubic function with a positive leading coefficient. - Cubic functions of the form $ax^3 + bx^2 + cx + d$ with $a \neq 0$ are generally one-to-one because they are strictly increasing or decreasing over their domain. - B. $f(x) = x^2 + x - 3$ - This is a quadratic function. - Quadratic functions are parabolas and fail the Horizontal Line Test, so they are not one-to-one. - C. $f(x) = |2x - 4|$ - Absolute value functions are V-shaped and not one-to-one because they map two different inputs to the same output. - D. $f(x) = x^3 - 2x - 1$ - This is a cubic function. - However, unlike the simple cubic in A, this function has a more complex shape and may not be strictly monotonic. - To check, find the derivative: $$f'(x) = 3x^2 - 2$$ - Set derivative to zero to find critical points: $$3x^2 - 2 = 0 \Rightarrow x^2 = \frac{2}{3} \Rightarrow x = \pm \sqrt{\frac{2}{3}}$$ - Since derivative changes sign, the function is not strictly increasing or decreasing, so it is not one-to-one. 4. **Conclusion:** Only function A is one-to-one. **Final answer:** Function A, $f(x) = 3x^3 + 1$, is one-to-one.