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
Multiple Choice Answers
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, cube root of, 9, x, minus, 8, end cube rootf
−1
(x)=
3

9x−8
​

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, cube root of, 9, x, end cube root, minus, 8f
−1
(x)=
3

9x
​
−8
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 9, cube root of, x, minus, 8, end cube rootf
−1
(x)=9
3

x−8
​

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 9, left parenthesis, cube root of, x, end cube root, minus, 8, right parenthesisf
−1
(x)=9(
3

x
​
−8)

Answer 1

To find the inverse function \( f^{-1}(x) \) of the given function \( f(x) = \frac{x^3 + 8}{9} \), we will follow the steps to isolate \( x \) in terms of \( y \).

1. Start with the equation: \[ y = \frac{x^3 + 8}{9} \]

2. Multiply both sides by 9 to eliminate the fraction: \[ 9y = x^3 + 8 \]

3. Subtract 8 from both sides: \[ 9y - 8 = x^3 \]

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

5. To express the inverse function, switch \( x \) and \( y \): \[ f^{-1}(x) = \sqrt[3]{9x - 8} \]

Now, let's look at the multiple-choice answers:

1. \( f^{-1}(x) = \sqrt[3]{9x - 8} \)
2. \( f^{-1}(x) = \sqrt[3]{9x} - 8 \)
3. \( f^{-1}(x) = 9\sqrt[3]{x} - 8 \)
4. \( f^{-1}(x) = 9\left(\sqrt[3]{x} - 8\right) \)

The correct answer is the first option:

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