Write the piecewise function in terms of unit step functions.

In summary, a piecewise function is a mathematical function that is defined by different equations or rules for different intervals of the input variable. It can be written in terms of other mathematical functions, but unit step functions are often used for their convenience and efficiency. Unit step functions have a value of 1 for a specific range of the input variable and 0 for all other values, and they help to define the different intervals and rules in a piecewise function. To write a piecewise function in terms of unit step functions, the different intervals and rules are determined and then combined using addition or subtraction. The purpose of using unit step functions in a piecewise function is to easily define different rules for different intervals and make the function more concise and manageable
  • #1
shamieh
539
0
Write the piecewise function
\[ f(t) = \begin{cases}
2t, & 0\leq t < 3 \\
6, & 3 \le t < 5 \\
2t, & t \ge 5 \\
\end{cases}
\]
in terms of unit step functions.

So here is what i;ve got just guessing , I don't think I'm correct. I really need some help. But I got:

$f(t) = 2t[u(t-0) - u(t-3)] + 6[u(t-3) - u(t-5)] + 2t[u(t-5) - u(t - \infty)]$

Which becomes

$f(t) = 2t[u(t) - u(t-3)] + 6[u(t-3) - u(t-5)] + 2t[u(t-5)]$
 
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  • #2
Looks good to me!
 

1. What is a piecewise function?

A piecewise function is a mathematical function that is defined by different equations or rules for different intervals of the input variable. It can be thought of as multiple functions "stitched" together to form one larger function.

2. What are unit step functions?

Unit step functions are mathematical functions that have a value of 1 for a specific range of the input variable, and 0 for all other values. They are often used in piecewise functions to define the different intervals and rules.

3. How do you write a piecewise function in terms of unit step functions?

To write a piecewise function in terms of unit step functions, you first determine the different intervals and rules for the function. Then, for each interval, you use a unit step function to define the rule for that interval. Finally, you combine all the unit step functions using addition or subtraction, depending on the specific rules.

4. What is the purpose of using unit step functions in a piecewise function?

Unit step functions allow us to easily define different rules for different intervals in a piecewise function. They also help to make the function more concise and easier to work with mathematically.

5. Can a piecewise function be written in terms of other mathematical functions instead of unit step functions?

Yes, piecewise functions can be written in terms of other mathematical functions such as absolute value, polynomial, or exponential functions. However, unit step functions are often the most convenient and efficient way to represent piecewise functions.

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