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NoblePenguin

Answer:

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

Step-by-step 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 [tex]\displaystyle{x=4y^3+8}[/tex].

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

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

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

thepingu

Answer:

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

Step-by-step 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 [tex]y=ax^b+c[/tex]

2. Swap where the x and y values are

3. Solve for y

4. Convert into [tex]f^{-1}(x)[/tex] form

So first, we can write [tex]f(x)=4x^3+8[/tex] as [tex]y=4x^3+8[/tex].

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

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

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

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

Hope this helped!

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