Verify that the functions  f(x)=  (x - 6)^(1/3) and g(x) = x^3 + 6 are inverse of each other.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You may use the alternative method to find what the inverse of the function g(x) is and to compare the inverse `g^(-1)(x)` to `f(x) ` function to check if they are alike.

Hence, you need to solve forx the equation `y = x^3 + 6`  such that:

`x^3 = y - 6 =gtx = root(3) (y-6)`

Hence, evaluating the inverse of the function g(x) yields `g^(-1)(x)=root(3) (x-6).`

Notice that the equation of the function `f(x) = (x-6)^(1/3),`  hence you may use radical sign to write the equation such that:

`f(x) = root(3) (x-6)`

Comparing the inverse of g(x) to the function f(x) yields that `g^(-1)(x) = f(x), ` hence the function f(x) and g(x) are inverse to each other.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function f(x) = `(x - 6)^(1/3)` and g(x) = x^3 + 6

fog(x) = f(g(x)) = f(x^3 + 6)

=> `(x^3 + 6 - 6)^(1/3)`

=> `(x^3)^(1/3) `

=> x

gof(x) = g(f(x))

=> `g((x - 6)^(1/3))`

=> `((x - 6)^(1/3))^3 + 6`

=> x - 6 + 6

=> x

This proves that f(x) = `(x - 6)^(1/3)` and g(x) = x^3 + 6 are inverse functions

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