Piecewise Function: Solving Options Assignment

In summary, the conversation discusses a formula for an Options assignment and the question of whether it can be rewritten as a single equation. One suggestion is to define a function on each interval and then combine them into one equation.
  • #1
vladimir69
130
0
hi,
i have been working on an Options assignment, and i have this formula as one of my answers:

[tex]X_{t}=\left\{\begin{array}{cc}X_{t-1},&\mbox{if } S_{t} < S_{t-1}\\aS_{t},& \mbox{if } S_{t}\geq S_{t-1}\end{array}\right[/tex]

where a is a constant. the question however asks for a "single equation", would the equation(s)? i have defined above be ok? is there a way to rewrite this not using a piecewise function. hopefully you know what i mean.

thnx
vladimir
 
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  • #2
Just a thought- find the definition of "equation" you were given. If you weren't given one...
As for another way to write a piecewise function, why not define a function on each of your intervals (the restriction of the piecewise function to each interval) and define the piecewise function as the union of those functions? I.e., given abs: R --> R,
[tex]\mbox{abs}(x) = \left\{\begin{array}{cc}x,&\mbox{if } x \geq 0 \\-x,& \mbox{if } x < 0 \end{array}\right[/tex]
Let f: R+ --> R, f(x) = x and g: R- --> R, g(x) = -x. Then abs = f U g - which is almost certainly a single equation ;) Doesn't save time, but I think it works.
 
Last edited:
  • #3


Hi Vladimir,

Great job on your Options assignment! Your formula looks correct and it is a valid piecewise function. However, if the question specifically asks for a "single equation", then you may need to rewrite it without using a piecewise function. Here are a few options you can explore:

1. Use absolute value notation: Instead of using the "if" condition, you can use absolute value notation to represent the two cases. Your equation would then be:

X_t = |aS_t| + |(1-a)S_{t-1}|

2. Use a conditional statement: You can also rewrite the equation using a conditional statement, such as "if-else" or "if-then-else". Your equation would then be:

X_t = \begin{cases} X_{t-1}, & \mbox{if } S_t < S_{t-1}\\ aS_t, & \mbox{if } S_t \geq S_{t-1} \end{cases}

3. Use a piecewise function with different cases: Another option is to rewrite the equation using a piecewise function with different cases, so that each case only has one condition. Your equation would then be:

X_t = \begin{cases} X_{t-1}, & \mbox{if } S_t < S_{t-1}\\ aS_t, & \mbox{if } S_t = S_{t-1}\\ aS_t + (1-a)S_{t-1}, & \mbox{if } S_t > S_{t-1} \end{cases}

Ultimately, the best way to rewrite the equation without using a piecewise function would depend on the specific requirements and context of your assignment. I hope these suggestions help and good luck with your assignment!
 

What is a piecewise function?

A piecewise function is a mathematical function that is defined by different sub-functions for different intervals or "pieces" of the domain. It is typically used to represent relationships that change at certain points or intervals.

How do you graph a piecewise function?

To graph a piecewise function, first identify the different sub-functions and their corresponding intervals. Then, plot the points for each sub-function on their respective intervals and connect them with a line. Be sure to indicate any breaks or discontinuities in the graph.

How do you solve a piecewise function?

To solve a piecewise function, you must first identify the domain and range of the function. Then, evaluate each sub-function for the given input values within their corresponding intervals. Finally, combine the results from each sub-function to get the final solution.

What are some real-life applications of piecewise functions?

Piecewise functions are commonly used in economics, finance, and engineering to model relationships that have different rules or behaviors at different points. One example is the cost of electricity, which may have different rates for different levels of usage. Another example is the depreciation of a car's value over time, which may have different rates for the first few years and then a constant rate after that.

How are piecewise functions related to continuity and differentiability?

Piecewise functions are not necessarily continuous or differentiable at all points. In order for a piecewise function to be continuous, the different sub-functions must connect smoothly at the points where they meet. For differentiability, the different sub-functions must be differentiable at the points where they meet and their derivatives must also match.

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