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

Answer:

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

Step-by-step 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 $\sqrt[3]{x}+2$