For the function f, of, x, equals, start fraction, x, cubed, plus, 8, divided by, 9, end fractionf(x)=

Question

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: \[ f^{-1}(x) = \sqrt[3]{9x - 8} \]