Question

Please help!!! I really need it

Answer 1

`\displaystyle{ {f}^( - 1) (x) = {(x - 4)}^(3) + 1}`

Explanation:

Let y = f(x)

`\displaystyle{y = \sqrt[3]{x - 1} + 4}`

Finding an inverse, swap x and y

`\displaystyle{x = \sqrt[3]{y- 1} + 4}`

Then solve for y, first subtract 4 both sides

`\displaystyle{ x - 4= \sqrt[3]{y- 1} + 4 - 4} \\ \\ \displaystyle{ x - 4= \sqrt[3]{y- 1} }`

Cube both sides

`\displaystyle{ {(x - 4)}^(3) = \left(\sqrt[3]{y- 1} \right)^(3) } \\ \\ \displaystyle{ {(x - 4)}^(3) = y - 1}`

Add both sides by 1

`\displaystyle{ {(x - 4)}^(3) + 1 = y - 1 + 1} \\ \\ \displaystyle{ {(x - 4)}^(3) + 1 = y}`

The final y is the inverse. Hence:

`\displaystyle{ {f}^( - 1) (x) = {(x - 4)}^(3) + 1}`