Question

7. Find the inverse of f(x) = 4x^3 + 8.

Answer 1

The inverse of
`\displaystyle{f(x)=4x^3+8}` is determined to be
`{f^(-1)(x)=\sqrt[3]{(1)/(4)x-8}`.

Explanation:

In order to find the inverse of a function, we need to know a couple of rules about functions.

• f(x) = y
• To find the inverse, we flip the y-variable with the x-variables and solve for y.

Following these rules, our function now becomes
`\displaystyle{x=4y^3+8}`.

We now can simplify it further in order to find the value of x.

`\displaystyle{x=4y^3+8}\\\\4y^3=x-8\\\\y^3=(x)/(4)-8\\\\\sqrt[3]{y^3}=\sqrt[3]{(x)/(4)-8} \\\\y = \sqrt[3]{(1)/(4)x-8}\\\\f^(-1)(x)= \sqrt[3]{(1)/(4)x-8}`

Therefore, the inverse of
`\displaystyle{f(x)=4x^3+8}` is
`\displaystyle{f^(-1)(x)=\sqrt[3]{(1)/(4)x-8}`.

Answer 2

`\huge\boxed{f^(-1)(x) = \sqrt[3]{(x-8)/(4)}}`

Explanation:

In order to find the inverse of a function, we need to follow a series of steps.

1. Write the function in the form
`y=ax^b+c`

2. Swap where the x and y values are

3. Solve for y

4. Convert into
`f^(-1)(x)` form

So first, we can write
`f(x)=4x^3+8` as
`y=4x^3+8`.

Now we need to swap where the x and y variables are. This makes our equation
`x=4y^3+8`.

To find the inverse, we now need to solve for y in this equation.

• 
  `x=4y^3+8`
• Subtract 8 from both sides:
• 
  `x-8 = 4y^3`
• 
  `4y^3 = x-8`
• Divide both sides by 4:
• 
  `y^3 = (x-8)/(4)`
• Take the cube root of each side:
• 
  `y = \sqrt[3]{(x-8)/(4)}`
• Convert into
  `f^(-1)(x)` form

• 
  `f^(-1)(x) = \sqrt[3]{(x-8)/(4)}`

Therefore, the inverse of this function is
`f^(-1)(x) = \sqrt[3]{(x-8)/(4)}`.

Hope this helped!