Rewriting a piecewise function using step functions

In summary, a piecewise function is a mathematical function that behaves differently depending on the value of its input. Step functions are a type of piecewise function that consists of constant values over different intervals, creating a step-like graph. To rewrite a piecewise function using step functions, it must be broken down into separate intervals and a step function must be defined for each interval. This approach has advantages such as making the function easier to understand and graph, but has limitations in terms of the number of intervals and potential inaccuracies in representing the function's behavior in between intervals.
  • #1
selig5560
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TL;DR Summary
Function properties
Suppose we have a piecewise function

f(t) = exp(c*t) when 0 <= t < 2 and f(t) = 0 when t >= 2.

Can the above be rewritten as

f(t)= exp(at)*[H(t-0) - H(t-2)],

H is a heaviside function.
 
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  • #2
What is ##a##?
 
  • #3
You didn't specify f(t) for t < 0.
 

1. How do you rewrite a piecewise function using step functions?

To rewrite a piecewise function using step functions, you need to first identify the different intervals or domains in which the function is defined. Then, for each interval, you can write a separate step function that represents the output for that particular interval. Finally, you can combine all the step functions to create the piecewise function.

2. What is a step function?

A step function is a function that changes its value suddenly at specific points, rather than continuously. It is also known as a staircase function or a piecewise constant function.

3. Why is it useful to rewrite a piecewise function using step functions?

Rewriting a piecewise function using step functions can make the function easier to work with and understand. It can also make it easier to graph the function and analyze its behavior, especially at the points where the function changes its value.

4. Can any piecewise function be rewritten using step functions?

Yes, any piecewise function can be rewritten using step functions. However, some piecewise functions may be simpler to work with in their original form, so it is not always necessary to rewrite them using step functions.

5. Are there any limitations to rewriting a piecewise function using step functions?

One limitation is that the step functions used to represent the different intervals must be continuous at the points where the function changes its value. This means that the left and right limits of the step functions must be equal at these points. Additionally, the step functions may not always accurately represent the original function, especially if the function has a lot of variability within each interval.

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