Question

For the function f, of, x, equals, start fraction, x, cubed, plus, 8, divided by, 9, end fractionf(x)=
9
x
3
+8
​
, find f, to the power minus 1 , left parenthesis, x, right parenthesisf
−1
(x)

Answer 1

To find the inverse function \( f^{-1}(x) \) for the function \( f(x) = \frac{x^3 + 8}{9} \), we will follow these steps:

1. **Set the function equal to \( y \):**
\[
y = \frac{x^3 + 8}{9}
\]

2. **Solve for \( x \) in terms of \( y \):**
First, we will multiply both sides by 9 to eliminate the fraction:
\[
9y = x^3 + 8
\]

Now, subtract 8 from both sides:
\[
x^3 = 9y - 8
\]

Finally, take the cube root of both sides to solve for \( x \):
\[
x = \sqrt[3]{9y - 8}
\]

3. **Swap \( x \) and \( y \) to find the inverse function:**
Now we replace \( y \) with \( x \) to express the inverse function:
\[
f^{-1}(x) = \sqrt[3]{9x - 8}
\]

So, the inverse function is:
\[
f^{-1}(x) = \sqrt[3]{9x - 8}
\]