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June 17th, 2018, 10:58 AM   #1
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How to solve for f(x)

Solve the equation, find the function f(x), where $\displaystyle f^{-1}(x) $ is inverse of f(x),
$\displaystyle f(x)=f^{-1}(x) $

Last edited by skipjack; June 17th, 2018 at 10:13 PM.
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June 17th, 2018, 11:53 AM   #2
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There is no single answer- there are many functions that are there own inverse. One obvious one is f(x)= x. Another is f(x)= 1/x.
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June 17th, 2018, 12:53 PM   #3
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$f(x) = -x$, which works in the real or complex numbers.

$f(z) = \overline{z}$, the complex conjugate of $z$.
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June 18th, 2018, 12:39 AM   #4
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$\displaystyle f(z) = \frac{a - bz}{b - cz}$, where $a$, $b$ and $c$ are constants that don't satisfy $ac = b^2\!$,
and $z$ doesn't satisfy $b - cz = 0$.
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June 19th, 2018, 09:51 AM   #5
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I post my work below:
$\displaystyle f(x)=f^{-1}f^{-1}(x)=f^{-1}(x) $
$\displaystyle f_1 ^{-1}(x)=x=f_1 (x) \; \; $ and $\displaystyle \; \; f_2 ^{-1}(x)=x$ so we got first solution $\displaystyle f_1 (x)=x$
Now I don't understand how to get the second solution $\displaystyle y=\frac{1}{x}$ from $\displaystyle f_2 ^{-1} (x) =x$

Last edited by skipjack; June 19th, 2018 at 10:13 AM.
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June 19th, 2018, 10:39 AM   #6
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Your first line, $f(x)=f^{-1}f^{-1}(x)=f^{-1}(x)$, isn't a consequence of the given information.

The given information implies that when the function is nested to an even depth, the result simplifies to $x$, and that when the function is nested to an odd depth, the result simplifies to $f(x)$.
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June 21st, 2018, 04:22 AM   #7
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I see but let me explain below :
Use $\displaystyle f(x)=f^{-1} f^{-1} (x)$ so equation now is $\displaystyle f^{-1}(x)=f^{-1} f^{-1} (x) $ or $\displaystyle f^{-1}(x)=x$

From the system of equations $\displaystyle \begin{cases} f(f^{-1} (x) )=f(x) \\ f(x)=f^{-1} (x) \end{cases} \Rightarrow f(f^{-1} (x))=f^{-1}(x) $
For $\displaystyle f^{-1}(x)=T $ then $\displaystyle f(t)=t $ which is the first solution
The second solution $\displaystyle y=1/x $ will take long write so im not posting it
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June 21st, 2018, 05:53 AM   #8
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Quote:
Originally Posted by idontknow View Post
How to solve for f(x)

Solve the equation
Done!
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