Finding Unknown Function f(x)

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In summary: Instead, the problem writer is giving you a function f(x) and asking you to show that it satisfies the given equation.
  • #1
fled143
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Homework Statement




f(x) = f(x-k) f(k) / [ cot(k) + cot(x-k) ]

Show that the solution of the equation is

f(x) = 1/sin(x)



Homework Equations



sin(-x) = -sin(x)
cot(x) = cos(x) / sin(x)



The Attempt at a Solution



Transform the cotangents into cos and sin and simplify

f(x) = f(k) f(x-k) sin(k) (-csc(x) ) sin(x-k) eqn(*)
 
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  • #2
All you have to do is substitute [itex]f(x)=csc(x)[/itex] into the equation and show that both sides are equal through simplication and use of trig identities.

[tex]f(x)=\frac{f(x-k)f(k)}{cot(k)+cot(x-k)}[/tex]

[itex]f(x)=csc(x)[/itex] and this means by its definition that [itex]f(x-k)=csc(x-k)[/itex] and [itex]f(k)=csc(k)[/itex]

Now you just need to show that [tex]csc(x)=\frac{csc(x-k)csc(k)}{cot(k)+cot(x-k)}[/tex]
 
  • #3
That is supposed to be easy assuming that I already know what the f(x) is. But actually the problem is that I need to derive the solution f(x) = csc(x) from the given equation. I'm sorry if I have not pose my problem clearly at the start.

Thanks for helping.
 
  • #4
The problem statement is somewhat ambiguous.
fled143 said:
Show that the solution of the equation is
f(x) = 1/sin(x)
One possible meaning for this sentence is that you need to show that the function f(x) = 1/sin(x) satisfies the given equation. In this case you are given that f(x) = 1/sin(x), which is also equal by definition to csc(x).

Another meaning that IMO is less likely is that you are supposed to solve the given equation and arrive at the solution f(x) = 1/sin(x). I don't believe that this is the intent of the problem. If that had been the case, the problem writer could have been clearer by asking you to solve the given equation for f(x).
 

What is the purpose of finding the unknown function f(x)?

The purpose of finding the unknown function f(x) is to understand the relationship between the input (x) and the output (f(x)) of a given system or data set. This can help us make predictions, identify patterns, and make informed decisions in various fields such as physics, biology, economics, and engineering.

How do you find the unknown function f(x)?

There are several methods for finding the unknown function f(x), including trial and error, curve fitting, regression analysis, and differential equations. These methods involve using mathematical equations and techniques to analyze the data and determine the most suitable function that represents the relationship between the input and output.

What are some real-world applications of finding the unknown function f(x)?

Finding the unknown function f(x) has various real-world applications, such as predicting the stock market, modeling population growth, designing efficient engines, and understanding the behavior of physical systems. It is also used in data analysis, machine learning, and artificial intelligence to make accurate predictions and decisions based on data.

What challenges are involved in finding the unknown function f(x)?

One of the main challenges in finding the unknown function f(x) is that there may not be a single function that accurately represents the relationship between the input and output. In such cases, multiple functions may need to be combined or more complex mathematical models may need to be used. Additionally, the data may be noisy or incomplete, making it difficult to determine the underlying relationship.

Can finding the unknown function f(x) guarantee accurate predictions?

No, finding the unknown function f(x) does not guarantee accurate predictions. The accuracy of predictions depends on the quality of the data, the chosen method for finding the unknown function, and the assumptions made in the process. It is important to carefully analyze and validate the results to ensure the predictions are as accurate as possible.

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