Math Problem Statement
g(t) = 8>>>><
: t; 0 t < 1 t + 1; 1 t < 2 t 2; 2 t < 3 0; t 3 (a) Express g(t) in terms of the unit step functions.
Solution
The function is given in piecewise form:
t & 0 \leq t < 1 \\ -t + 1 & 1 \leq t < 2 \\ t - 2 & 2 \leq t < 3 \\ 0 & t \geq 3 \end{cases}$$ To express this function in terms of unit step functions, we use the following general idea: - The unit step function $$u(t)$$ is defined as: $$u(t) = \begin{cases} 0 & t < 0 \\ 1 & t \geq 0 \end{cases}$$ - For each segment of the piecewise function, we need to shift and scale the unit step function appropriately. ### Breaking down the function: 1. **For $$0 \leq t < 1$$, $$g(t) = t$$:** - This segment starts at $$t = 0$$ and ends at $$t = 1$$. To express this using unit step functions, we use: $$t \cdot \left[ u(t) - u(t-1) \right]$$ Here, $$u(t)$$ activates the function starting from $$t = 0$$, and $$u(t-1)$$ turns it off at $$t = 1$$. 2. **For $$1 \leq t < 2$$, $$g(t) = -t + 1$$:** - This segment starts at $$t = 1$$ and ends at $$t = 2$$. We express this as: $$(-t + 1) \cdot \left[ u(t-1) - u(t-2) \right]$$ Here, $$u(t-1)$$ starts the function at $$t = 1$$, and $$u(t-2)$$ ends it at $$t = 2$$. 3. **For $$2 \leq t < 3$$, $$g(t) = t - 2$$:** - This segment starts at $$t = 2$$ and ends at $$t = 3$$. We express it as: $$(t - 2) \cdot \left[ u(t-2) - u(t-3) \right]$$ The step function $$u(t-2)$$ activates the segment at $$t = 2$$, and $$u(t-3)$$ ends it at $$t = 3$$. 4. **For $$t \geq 3$$, $$g(t) = 0$$:** - There is no contribution from this segment since the function is zero for $$t \geq 3$$. We can ignore it in the expression. ### Combining all terms: The final expression for $$g(t)$$ in terms of unit step functions is: \[ g(t) = t \cdot \left[ u(t) - u(t-1) \right] + (-t + 1) \cdot \left[ u(t-1) - u(t-2) \right] + (t - 2) \cdot \left[ u(t-2) - u(t-3) \right] \] ### Final answer: \[ g(t) = t \cdot \left[ u(t) - u(t-1) \right] + (-t + 1) \cdot \left[ u(t-1) - u(t-2) \right] + (t - 2) \cdot \left[ u(t-2) - u(t-3) \right] \] --- Would you like to see further details or have any specific questions about the steps? Here are some related questions: 1. How do unit step functions help in simplifying piecewise functions? 2. What are the properties of the unit step function? 3. How would you solve integrals involving unit step functions? 4. What are the applications of using unit step functions in signal processing? 5. How would you handle piecewise functions that involve other operations like multiplication or division? **Tip:** Always break down each piece of the piecewise function and apply the unit step function separately for each interval!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Unit Step Functions
Function Representation
Formulas
g(t) = t [u(t) - u(t-1)]
g(t) = (-t + 1) [u(t-1) - u(t-2)]
g(t) = (t - 2) [u(t-2) - u(t-3)]
Theorems
Unit Step Function Representation of Piecewise Functions
Suitable Grade Level
Grades 11-12
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