Question

I need to verify identity functions for a one to one function.

Answer 1

We have the function

`f(x)=(x+6)^3`

1. For f^-1:

Let y = f(x) = (x+6)^3

Switch x and y to get:

`x=(y+6)^3`

And solve for y

`\begin{gathered} x^{(1)/(3)}=y+6 \\ x^{(1)/(3)}-6=y+6-6 \\ x^{(1)/(3)}-6=y \end{gathered}`

And we have y = f^-1(x)

Answer blank 1:

`f^(-1)(x)=x^{(1)/(3)}-6`

2. For f o f^-1 (x):

`(f\circ f^(-1))(x)=f(f^(-1)(x))`

And solve

`\begin{gathered} =f(x^{(1)/(3)}-6) \\ =(x^{(1)/(3)}-6+6)^3 \\ =(x^{(1)/(3)})^3 \\ =x \end{gathered}`

answer blank 2

`x^{(1)/(3)}-6`

answer blank 3

`x^{(1)/(3)}-6`

answer blank 4

`x^{(1)/(3)}`

3. For f^-1 o f:

`(f^(-1)\circ f)(x)=f^(-1)(f(x))`

Solve

`\begin{gathered} =f^(-1)((x+6)^3) \\ =\sqrt[3]{(x+6)^3}-6 \\ =x+6-6 \\ =x \end{gathered}`

answer blank 5

`(x+6)^3`

answer blank 6

`(x+6)^3`

answer blank 7

`x+6`