How would you find #f^-1# and verify that #f(f^-1(x))=f^-1(f(x))=x#, if #f(x)=2x+3#? #f(x)=x^3+1#?

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Answer 1 · 2016-08-04T16:16:29

#f(f^(-1)(x))= f^(-1)(f(x))#, if and only if f(x)=x.

Answer 2 · 2016-08-12T21:20:45

I'll do #f(x) = x^3 + 1# and leave #f(x) = 2x + 3# up to you to do for practice.

The inverse of a function can be found algebraically by switching the values of #x# and #y# inside the function:

#y = x^3 + 1#

#x = y^3 + 1#

#x - 1 = y^3#

#root(3)(x -1) = y#

Let's now use your trick that this new function is indeed the inverse function.

#(root(3)(x - 1))^3 + 1 =^? x#

#x - 1 + 1 =^? x#

#x = x#

So, if #f(x) = x^3 + 1#, then #f^-1(x) = root(3)(x - 1)#

Hopefully this helps!