Question

Use algebra to find the inverse of the function f(x)=3x^5+7

Answer 1

The inverse of the function is
`f^(-1)(x) =\sqrt[5]{(x-7)/(3)}`

Explanation:

The steps to get the inverse of a function are:

1. Replace f(x) with y
2. Replace every x by y and every y by x
3. Solve the new equation from for y
4. Replace y with
   `f^(-1)(x)`

∵
`f(x)=3x^(5)+7`

- Replace f(x) by y

∴
`y=3x^(5)+7`

- Replace every x by y and every y by x

∴
`x=3y^(5)+7`

- Subtract 7 from both sides

∴
`x-7=3y^(5)`

- Divide both sides by 3

∴
`(x-7)/(3)=y^(5)`

- Insert fifth root for both sides

∴
`\sqrt[5]{(x-7)/(3)}=y`

- Switch the two sides

∴
`y=\sqrt[5]{(x-7)/(3)}`

- replace y by
`f^(-1)(x)`

∴
`f^(-1)(x) =\sqrt[5]{(x-7)/(3)}`

The inverse of the function is
`f^(-1)(x) =\sqrt[5]{(x-7)/(3)}`