How do you find the inverse of #f(x)=root3(x+2)#?

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Answer 1 · 2017-01-30T19:54:27

Substitute #f^-1(x)# for x
Use the property #f(f^-1(x)) = x#
Solve for #f^-1(x)#

Substitute #f^-1(x)# for x

#f(f^-1(x)) = root(3)(f^-1(x)+2)#

Use the property #f(f^-1(x)) = x#

#x = root(3)(f^-1(x)+2)#

Cube both sides:

#x^3 = (root(3)(f^-1(x)+2))^3#

The cube and the cube root are inverses so they disappear:

#x^3 = f^-1(x)+2#

Subtract 2 from both sides:

#f^-1(x) = x^3-2#

Verify that #f(f^-1(x)) = x#

#f(f^-1(x)) = root(3)((x^3-2)+2)#

#f(f^-1(x)) = root(3)(x^3)#

#f(f^-1(x)) = x" "larr# verified

Vertify that #f^-1(f(x)) = x#:

#f^-1(f(x)) = (root(3)(x+2))^3-2#

#f^-1(f(x)) = x+2-2#

#f^-1(f(x)) = x" "larr# verified

Because we have verified #f(f^-1(x)) = f^-1(f(x)) = x#, then we can declare that:

#f^-1(x) = x^3-2#