Question 1209327
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Let f(x) = (2x + 5)/(x - 4).
If f^{-1} is the inverse of f, what is f^{-1}(1)?
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<pre>
All you need to do to find f^{-1}(1)  is to solve this equation for x

    {{{(2x+5)/(x-4)}}} = 1.    <<<---===  the sign is corrected after the notice from @greenestamp.


It can be solved in a few lines

    2x + 5 = x - 4.

    2x - x = -4 - 5

       x   =    -9.     


<U>ANSWEER</U>.  f^{-1}(1) is -9.
</pre>

Solved.


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What I am trying to explain is that in this problem you do not need
to restore the inverse function &nbsp;f^{-1}(x) &nbsp;explicitly as a function of &nbsp;x,
as the other tutor does. &nbsp;&nbsp;It is unnecessary work.


All you need to do to find &nbsp;f^{-1}(1) &nbsp;is to solve this equation, &nbsp;f(x) = 1, &nbsp;for &nbsp;x.


In problems of this kind, &nbsp;it is necessary to restore &nbsp;f^(-1)(x) &nbsp;explicitly in two cases: 


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1) &nbsp;&nbsp;if the problem explicitly asks you about it, 


&nbsp;&nbsp;&nbsp;&nbsp;and/or


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2) &nbsp;&nbsp;if the problem asks to calculate &nbsp;f^(-1)(x) &nbsp;for several/many values of x.



Then restoring the inverse function is justified from the point of view of effectiveness of your efforts.


Otherwise, &nbsp;if you are asked to find &nbsp;f^(-1)(x) &nbsp;for one single value of &nbsp;x = c,
it is more effective to solve an equation &nbsp;f(x) = c &nbsp;for this single value of &nbsp;"c".