1. **Stating the problem:** We have a piecewise function for position $x(t)$ defined over three time intervals: - For $0 < t < 1$ seconds, $x(t) = 10t$ meters. - For $1 < t \leq 4$ seconds, $x(t) = 38 + 2t^2 - 3t^{3/4}$ meters. - For $4 < t < 5$ seconds, $x(t) = 2t + 20$ meters. We want to understand and analyze this function. 2. **Formula and rules:** The function is piecewise, meaning it has different expressions depending on the time interval. We can evaluate position at any time $t$ by using the correct formula for that interval. 3. **Intermediate work:** - For $0 < t < 1$, position is linear: $x(t) = 10t$. - For $1 < t \leq 4$, position is a combination of polynomial and fractional powers: $x(t) = 38 + 2t^2 - 3t^{3/4}$. - For $4 < t < 5$, position is again linear: $x(t) = 2t + 20$. 4. **Example evaluations:** - At $t=0.5$ (in first interval): $x(0.5) = 10 \times 0.5 = 5$ meters. - At $t=2$ (in second interval): $x(2) = 38 + 2 \times 2^2 - 3 \times 2^{3/4} = 38 + 8 - 3 \times 1.6818 \approx 38 + 8 - 5.045 = 40.955$ meters. - At $t=4.5$ (in third interval): $x(4.5) = 2 \times 4.5 + 20 = 9 + 20 = 29$ meters. 5. **Explanation:** This function models position over time with different behaviors in each interval. The first and last intervals are linear motions, while the middle interval has a more complex motion involving quadratic and fractional powers. Final answer: The piecewise function is correctly defined as given, and can be evaluated at any $t$ in the specified intervals using the corresponding formula.