Question 1194862
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You can find the inverse of many relatively simple functions without having to switch the x and y and solve for the new y, as shown by the other tutor.<br>
When you work the problem that way, after you switch the x and y the operations you need to perform to solve for the new y are (1) subtract 2, (2) take the cube root, and (3) add 1 -- leading to the inverse function {{{y=root(3,x-2)+1}}}<br>
You can find that inverse without doing the algebra by using the concept that the inverse function "gets you back where you started".<br>
An inverse function, to "get you back where you started", has to perform the opposite operations, and in the opposite order, compared to the given function.<br>
The given function performs the following sequence of operations on the input value:
(1) subtract 1
(2) raise to power 3
(3) add 2<br>
The inverse function therefore needs to perform the following sequence of operations:
(1) subtract 2
(2) take the cube root
(3) add 1<br>
which gives the inverse function as
{{{x}}} --> {{{x-2}}} --> {{{root(3,x-2)}}} --> {{{root(3,x-2)+1}}}<br>
The steps you perform in forming this inverse function are exactly the steps you need to perform if you switch the x and y and solve for the new y -- but you don't need to do all that algebra.<br>