SOLUTION: Let 𝑓 be the function with correspondence rule 𝑓(𝑥) = (𝑥 − 1) ^3+ 2, determine the inverse function 𝑓^−1 and its domain.

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Let 𝑓 be the function with correspondence rule 𝑓(𝑥) = (𝑥 − 1) ^3+ 2, determine the inverse function 𝑓^−1 and its domain.      Log On


   

Question 1194862: Let 𝑓 be the function with correspondence rule 𝑓(𝑥) = (𝑥 − 1) ^3+ 2, determine the inverse function 𝑓^−1
and its domain.
Found 3 solutions by MathLover1, MathTherapy, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!


if f%28x%29+=+%28x+-+1%29%5E3%2B+2, then
f%28x%29+=y
y=+%28x+-+1%29%5E3%2B+2.........swap variables
x=+%28y+-+1%29%5E3%2B+2...........solve for y
x-2=+%28y+-+1%29%5E3+
root%283%2Cx-2%29=y-1
y=root%283%2Cx-2%29%2B1
f%5E-1+=root%283%2Cx-2%29%2B1
domain: { x element R : x%3E=2 }


desmos-graph-14

Answer by MathTherapy(10556) About Me  (Show Source):
You can put this solution on YOUR website!
Let 𝑓 be the function with correspondence rule 𝑓(𝑥) = (𝑥 − 1) ^3+ 2, determine the inverse function 𝑓^−1
and its domain.
Whether you're asking for the DOMAIN of the given function or of its inverse,

the domain is the same, ALL REALS!!

Answer by greenestamps(13206) About Me  (Show Source):
You can put this solution on YOUR website!


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.

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

You can find that inverse without doing the algebra by using the concept that the inverse function "gets you back where you started".

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.

The given function performs the following sequence of operations on the input value:
(1) subtract 1
(2) raise to power 3
(3) add 2

The inverse function therefore needs to perform the following sequence of operations:
(1) subtract 2
(2) take the cube root
(3) add 1

which gives the inverse function as
x --> x-2 --> root%283%2Cx-2%29 --> root%283%2Cx-2%29%2B1

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.