document.write( "Question 1194862: Let 𝑓 be the function with correspondence rule 𝑓(𝑥) = (𝑥 − 1) ^3+ 2, determine the inverse function 𝑓^−1
\n" ); document.write( "and its domain. \n" ); document.write( "

Algebra.Com's Answer #827149 by greenestamps(13206) $\"\"$ $\"About$

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\n" ); document.write( "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.

\n" ); document.write( "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 $\"y=root%283%2Cx-2%29%2B1\"$

\n" ); document.write( "You can find that inverse without doing the algebra by using the concept that the inverse function \"gets you back where you started\".

\n" ); document.write( "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.

\n" ); document.write( "The given function performs the following sequence of operations on the input value:
\n" ); document.write( "(1) subtract 1
\n" ); document.write( "(2) raise to power 3
\n" ); document.write( "(3) add 2

\n" ); document.write( "The inverse function therefore needs to perform the following sequence of operations:
\n" ); document.write( "(1) subtract 2
\n" ); document.write( "(2) take the cube root
\n" ); document.write( "(3) add 1

\n" ); document.write( "which gives the inverse function as
\n" ); document.write( " $\"x\"$ --> $\"x-2\"$ --> $\"root%283%2Cx-2%29\"$ --> $\"root%283%2Cx-2%29%2B1\"$

\n" ); document.write( "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.

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