Find the inverse of f(x) in 4 minutes

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In summary, the conversation discusses the topic of time limits and the function $y=\frac{e^x-e^{-x}}{e^x+e^{-x}}$, which is equivalent to $\tanh x$. It also mentions the inverse hyperbolic tangent function and its representation as $\frac{1}{2}\ln\left(\frac{x+1}{1-x}\right)$.
  • #1
Lorena_Santoro
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  • #2
why a time limit?

$y=\dfrac{e^x-e^{-x}}{e^x+e^{-x}} = \dfrac{e^{2x}-1}{e^{2x}+1}$

$x = \dfrac{e^{2y}-1}{e^{2y}+1}$

$xe^{2y}+x = e^{2y}-1$

$x+1=e^{2y}(1-x)$

$e^{2y}=\dfrac{x+1}{1-x} \implies y = \dfrac{1}{2}\ln\left(\dfrac{x+1}{1-x}\right)$
 
  • #3
It's a math-quiz like channel.
 
  • #4
We might recognize the definition of $\sinh x=\frac 12(e^x-e^{-x})$ and its associates $\cosh x$ and $\tanh x$.
We have $\tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$, which is the same as the given $f(x)$.
Therefore $f^{-1}(x)=\tanh^{-1} x=\artanh x$, which happens to be the same as $\frac 12\ln\left(\frac{1+x}{1-x}\right)$ as skeeter showed. See Inverse hyperbolic tangent on wiki.
 
  • #5
What should I do with the other three minutes?
 
  • #6
Country Boy said:
What should I do with the other three minutes?
Enjoy being so smart! ;-)
 
  • #7
And a nice cup of Earl Grey tea!
 
  • #8
I'd totally agree!
 

1. What is the purpose of finding the inverse of a function?

The purpose of finding the inverse of a function is to be able to undo the original function and solve for the input value (x) given an output value (y). This allows for easier problem solving and can also help in graphing and understanding the behavior of a function.

2. How do you find the inverse of a function?

To find the inverse of a function, you can follow these steps:

  1. Write the original function in the form of y = f(x)
  2. Switch the positions of x and y, so it becomes x = f(y)
  3. Solve for y by isolating it on one side of the equation
  4. The resulting equation is the inverse of the original function, written as y = f-1(x)

3. Can every function have an inverse?

No, not every function has an inverse. For a function to have an inverse, it must pass the horizontal line test, meaning that every horizontal line intersects the function at most once. If a function fails this test, it does not have an inverse.

4. Are there any special cases when finding the inverse of a function?

Yes, there are two special cases when finding the inverse of a function:

  • If the function is a horizontal line, it does not have an inverse
  • If the function is a vertical line, the inverse is not a function

5. How long does it typically take to find the inverse of a function?

The time it takes to find the inverse of a function can vary depending on the complexity of the function and the method used to find the inverse. However, with practice and familiarity with the process, it can typically be done in a few minutes.

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