1. **Problem:** Find the function expressions for the given intervals. 2. **Step 1:** For $f(t) = t$, $0 < t < \pi$. - This is a simple linear function increasing from 0 to $\pi$. 3. **Step 2:** For $f(t) = \pi - t$, $0 < t < \pi$. - This is a linear function decreasing from $\pi$ to 0. 4. **Step 3:** For $f(t) = t(a - t)$, $0 < t < a$. - This is a quadratic function with roots at $t=0$ and $t=a$. 5. **Step 4:** For $f(t) = e^{kt}$, $0 < t < a$. - This is an exponential growth or decay function depending on the sign of $k$. 6. **Step 5:** For piecewise function: $$f(t) = \begin{cases} t, & 0 \leq t \leq \frac{1}{2} \\ 1 - t, & \frac{1}{2} \leq t \leq 1 \end{cases}$$ - This is a linear increase then decrease forming a triangle shape. 7. **Step 6:** For piecewise function: $$f(t) = \begin{cases} t, & 0 < t \leq 1 \\ 1, & 1 < t < 2 \end{cases}$$ - Linear increase from 0 to 1, then constant 1. 8. **Step 7:** For $f(t) = \pi^2 - t^2$, $0 < t < \pi$. - This is a downward opening parabola with vertex at $t=0$. 9. **Step 8:** For piecewise function: $$f(t) = \begin{cases} 0, & 0 < t < \frac{a}{2} \\ 1, & \frac{a}{2} < t < a \end{cases}$$ - Step function jumping from 0 to 1 at $t=\frac{a}{2}$. **Summary:** Each function is defined clearly on its interval with linear, quadratic, exponential, or piecewise forms as given. Final answer: Functions 1 to 8 are as stated above with their respective intervals and expressions.