Question

what is the inverse of (x+2)^3

Need ASAP Will medal

Answer 1

solve \[x=(y+2)^3\]for \(y\) in two steps

Answer 2

a) take the cubed root of both sides
b) subtract 2 from both sides

Answer 3

In general (for odd integer \(n\)).
I added a b coefficient, or else it would have been too explicit, although it already is pretty simple.

Suppose you want to find the inverse of the following function,

\(\color{#000000 }{ \displaystyle f(x)=b(x+a)^n }\)

■ replacing f(x), by y for convinience

\(\color{#000000 }{ \displaystyle y=b(x+a)^n }\)

■ swiping the x and the y

\(\color{#000000 }{ \displaystyle x=b(y+a)^n }\)

■ swiping the x and the y

\(\color{#000000 }{ \displaystyle x=b(y+a)^n }\)

■ solve for y.

\(\color{#000000 }{ \displaystyle \sqrt[n]{x}=\sqrt[n]{b(y+a)^n} }\)

\(\color{#000000 }{ \displaystyle \sqrt[n]{x}=\sqrt[n]{~b~}\cdot \sqrt[n]{(y+a)^n} }\)

\(\color{#000000 }{ \displaystyle \sqrt[n]{x}=\sqrt[n]{~b~}(y+a) }\)

\(\color{#000000 }{ \displaystyle \sqrt[n]{x}=y\sqrt[n]{~b~}+a\sqrt[n]{~b~} }\)

\(\color{#000000 }{ \displaystyle \sqrt[n]{x}-a\sqrt[n]{~b~}=y\sqrt[n]{~b~} }\)

\(\color{#000000 }{ \displaystyle \frac{\sqrt[n]{x}-a\sqrt[n]{~b~}}{\sqrt[n]{~b~}}=y }\)

■ since y is an inverse function, it has a notation \(\).

\(\color{#000000 }{ \displaystyle f^{-1}(x)=\frac{\sqrt[n]{x}-a\sqrt[n]{~b~}}{\sqrt[n]{~b~}} }\)