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Problems for "Inverse Functions"

Problems for "Inverse Functions"

Problems for "Inverse Functions"

Problems for "Inverse Functions"

Problems for "Inverse Functions"

Problems for "Inverse Functions"

Problem : Find the inverse of f (x) = .

y =    

Solve for x first:


(x - 1)y   = x + 2  
xy - y   = x + 2  
xy - x   = y + 2  
x   =  

Now switch the variables x and y :

y =    

Problem : Find the inverse of f (x) = 6 + 5x 3 .


y   = 6 + 5x 3  
x   =  

Now switch variables:

y =    

Problem : If f (3) = 2 and f'(3) = 7 , what is (f -1)'(2) ?

(f -1)(2) = , since the slope of the inverse is the reciprocal.

Problem : Find (f -1)'(2) for f (x) = 4x 3 - 2x + 2 .

Note that f is not one-to-one throughout its domain, so it does not have an inverse defined on its entire range. However, there is a unique x , namely 0, such that f (x) = 2 . So, for a suitable domain containing 0, the inverse can be defined, and we may compute:


(f -1)'(2)   =  
    =  
    = -  

Problem : Find (f -1)'(- 4) for f (x) = x 3 - x 2 - 4x .

This problem doesn't make much sense, since there are several x 's such that f (x) = - 4 . Namely, they are x = - 2 , 1 , or 2 .