Question

Which function is the inverse of f(x)= x^3-6x^2+12x-8

Answer 1

We're not given the choices but we don't need them.

The only way this problem is tractable to a middle or high schooler is if the polynomial is perfect cube; a little thought yields

`f(x) = (x-2)^3`

To find the inverse let's call f(x) x and call x y and solve for y.

`x = (y-2)^3`

`y-2 = \sqrt[3]{x}`

`y = 2 + \sqrt[3]{x}`

Answer 2

`f^(-1)(x)=\sqrt[3]{x}+2`

Explanation:

Given:
`f(x)=x^3-6x^2+12x-8`

We need to find inverse of the f(x).

It is cubic equation. First we make perfect cube of f(x).

`a^3-3a^2b+3ab^2-b^3=(a-b)^3`

`x^3-6x^2+12x-8\Rightarrow x^3-3\cdot x^2\cdot 2+3\cdot x\cdot 2^2-2^3`

`a\rightarrow x`

`b\rightarrow 2`

`f(x)=(x-2)^3`

Now, we find the inverse of f(x).

Step 1: set f(x)=y

`y=(x-2)^3`

Step 2: switch x and y

`x=(y-2)^3`

Step 3: solve for y

`\sqrt[3]{x}=y-2`

`y=\sqrt[3]{x}+2`

Hence, The inverse of f(x) is
`\sqrt[3]{x}+2`