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

User Mpeerman
by
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2 Answers

3 votes

Answer:

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}.

User FDuhen
by
8.7k points
2 votes

Answer:


\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!

User Void Spirit
by
8.2k points

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