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Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. The function often takes the form of \( f(x) = a \cdot e^{bx} \), where \( a \) and \( b \) are constants, and \( e \) is Euler's number, approximately equal to 2.718. One of the properties of exponential functions is their rate of growth or decay.

- **Growth**: When \( b > 0 \), the function is increasing rapidly, representing exponential growth.
- **Decay**: When \( b < 0 \), such as in \( e^{-t} \), the function is decreasing rapidly, indicating exponential decay.

In the given functions, \( f(t) = e^{-t} \) showcases exponential decay, where the function decreases as the value of \( t \) increases.

Graph sketching involves plotting a function's behavior on a coordinate plane. It helps to visualize how the function behaves across different values of \( t \). Understanding the effects of the Heaviside step function on the graph is crucial in this exercise.

- **Function behavior**: For (a) \( f(t) = H(t-1) \cdot e^{-t} \), the step function \( H(t-1) \) ensures that the graph exists only for \( t \geq 1 \), starting at \( e^{-1} \) and decreasing thereafter. For (b) \( f(t) = [H(t-1) - H(t-3)] \cdot e^{-t} \), the graph is active between \( t=1 \) and \( t=3 \).

- **Steps to sketch**: - Mark the crucial points dictated by the step function on the \( t \)-axis. - Between these points, sketch the section of the exponential decay curve. - Return to \( y=0 \) outside the intervals defined by the Heaviside function.

This method helps in accurately visualizing the behavior of such piecewise functions linked with exponential decay.

Mathematical analysis involves a deeper investigation of function characteristics and behaviors. In this exercise, the interaction between exponential functions and Heaviside step functions exemplifies key ideas in analysis.

- **Heaviside Step Function**: A crucial concept here, used to "turn on" the graph of a function at certain points. In essence, for \( f(t) = H(t-a)\cdot g(t) \), the value of \( g(t) \) is zero before \( t = a \) and follows its usual behavior beyond that point.
- **Boundaries and Intervals**: Identifying these is vital in analyzing the input that affects the output of the function. In (b)'s example, the interval \( 1 \leq t < 3 \) signifies the region where the function is activated, impacting continuity and differentiability analysis.

Understanding these concepts helps frame a more comprehensive strategy when confronting real-world problems that use similar mathematical structures involving control functions like the Heaviside step.