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

The correct answer is the first option:

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