1. **State the problem:** We are given the function $$S(t) = \frac{1}{2}gt^2 + at + b$$ where $g$ is the acceleration due to gravity, and $a$, $b$ are constants. We need to check if this function is one-to-one (injective). 2. **Recall the definition of one-to-one:** A function is one-to-one if for every pair of distinct inputs $t_1$ and $t_2$, the outputs are distinct, i.e., if $S(t_1) = S(t_2)$ then $t_1 = t_2$. 3. **Analyze the function:** The function is a quadratic polynomial in $t$ with leading coefficient $\frac{1}{2}g$. Since $g$ (acceleration due to gravity) is positive (approximately 9.8), the parabola opens upwards. 4. **Check the derivative:** The derivative is $$S'(t) = gt + a$$ which is a linear function. 5. **Determine monotonicity:** The function $S(t)$ is increasing where $S'(t) > 0$ and decreasing where $S'(t) < 0$. Since $S'(t)$ is linear, it changes sign at $t = -\frac{a}{g}$. 6. **Conclusion on one-to-one:** Because $S(t)$ is a parabola opening upwards, it is not strictly increasing or decreasing over all real numbers. It decreases before $t = -\frac{a}{g}$ and increases after. Therefore, there exist distinct $t_1$ and $t_2$ such that $S(t_1) = S(t_2)$. Hence, the function $S(t)$ is **not one-to-one** over the entire real line. **Final answer:** The function $S(t) = \frac{1}{2}gt^2 + at + b$ is not one-to-one.