Question

Find the inverse of the function. f(x) = the cube root of quantity x divided by seven. - 9

Answer 1

to solve

replace f(x) with y
switch x and y
solve for y
replace y with f⁻¹(x)

so
I'm not sure if it is

`f(x)=\sqrt[3]{(x)/(7)}-9` or

`f(x)=\sqrt[3]{(x)/(7)-9}`


first one

`f(x)=\sqrt[3]{(x)/(7)}-9`
replace f(x) with y

`y=\sqrt[3]{(x)/(7)}-9`
switch x and y

`x=\sqrt[3]{(y)/(7)}-9`
solve for y

`x+9=\sqrt[3]{(y)/(7)}`

`(x+9)^3=(y)/(7)`

`7(x+9)^3=y`
replace y with f⁻¹(x)

`f^(-1)(x)=7(x+9)^3`

2nd one

`f(x)=\sqrt[3]{(x)/(7)-9}`
replce f(x) with y

`y=\sqrt[3]{(x)/(7)-9}`
switch x and y

`x=\sqrt[3]{(y)/(7)-9}`
solve for y

`x^3=(y)/(7)-9`

`x^3+9=(y)/(7)`

`7(x^3+9)=y`
replace y with f⁻¹(x)

`f^(-1)(x)=7(x^3+9)`



if you meant
`f(x)=\sqrt[3]{(x)/(7)}-9` then the inverse is
`f^(-1)(x)=7(x+9)^3`

if you meant
`f(x)=\sqrt[3]{(x)/(7)-9}` then the inverse is
`f^(-1)(x)=7(x^3+9)`