Question

Find, then state if it’s a function

Answer 1

The answer is y = ±
`\sqrt[4]{(x)/(4)}` ,
`f^(-1)(x)` is a function ⇒ 1st answer

Explanation:

Let us revise the steps of find the inverse of a function

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

∵
`f(x)=4x^(4)`

- Replace f(x) by y

∴
`y=4x^(4)`

- Replace y by x and x by y

∴
`x=4y^(4)`

- Divide each side by 4

∴
`(x)/(4)=y^(4)`

- Take
`\sqrt[4]{}` for both sides

∴ ±
`\sqrt[4]{(x)/(4)}=y`

- Switch the two sides

∴ y = ±
`\sqrt[4]{(x)/(4)}`

∵ There is no fourth root for negative number

∴ x ≥ 0

When you test the graph of
`f^(-1)(x)` by a vertical line, it will cut it just at one in every position, so it is a function. Look to the attached graph for more understand

∴
`f^(-1)(x)` is a function

The answer is y = ±
`\sqrt[4]{(x)/(4)}` ,
`f^(-1)(x)` is a function