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

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DeanR DeanR · Answer 1 · 2018-07-18T05:02:25+02:00

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

[tex]f(x) = (x-2)^3[/tex]

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

[tex]x = (y-2)^3[/tex]

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

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

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isyllus isyllus · Answer 2 · 2019-06-20T07:46:16+02:00

Answer:

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

Step-by-step explanation:

Given: [tex]f(x)=x^3-6x^2+12x-8[/tex]

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

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

[tex]a^3-3a^2b+3ab^2-b^3=(a-b)^3[/tex]

[tex]x^3-6x^2+12x-8\Rightarrow x^3-3\cdot x^2\cdot 2+3\cdot x\cdot 2^2-2^3[/tex]

[tex]a\rightarrow x[/tex]

[tex]b\rightarrow 2[/tex]

[tex]f(x)=(x-2)^3[/tex]

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

Step 1: set f(x)=y

[tex]y=(x-2)^3[/tex]

Step 2: switch x and y

[tex]x=(y-2)^3[/tex]

Step 3: solve for y

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

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

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