Question

IM TAKING AN EXAM AND I NEED SO MUCH HELP!!

Show how to find the inverse of f(x) = x^3 - 5. Calculate 3 points on f(x) and use these points to show that the inverse is correct.

Answer 1

`f^(-1)(x) = \sqrt[3]{x+5}`

Explanation:

A function
`f^(-1)` is the inverse of f if whenever y =f(x) and
`x =f^(-1)`

To find the inverse of f(x) =
`x^(3)` - 5, we use the following steps:

• Step 1 : Put y for f(x) and solve for x

y =
`x^(3)` - 5

<=>
`x^(3)` = y + 5

<=> x =
`\sqrt[3]{y+ 5}`

• Step 2: Put
  `f^(-1)(y)` for x, we have:

`f^(-1)(y) =\sqrt[3]{y+5}`

• Step 3: Interchange y =x, we have

`f^(-1)(x) = \sqrt[3]{x+5}`

Let Calculate 3 points on f(x)

x = 0 => y = -5

x = 1 => y = -4

x = 2 => y = 3

Let Calculate 3 points on
`f^(-1)` (x)

x = -5 => y = 0

x = -4 => y = 1

x = 3 => y = 2

Yes, the inverse is correct because:

• the domain of the inverse function is the range of the original function

• the range of the inverse function is the domain of the original function