For the function f, of, x, equals, 8, cube root of, x, end cube rootf(x)=8

Question

3

x
​
, 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, left parenthesis, 8, x, right parenthesis, cubedf
−1
(x)=(8x)
3

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, cube root of, left parenthesis, start fraction, x, divided by, 8, end fraction, right parenthesis, end cube rootf
−1
(x)=
3

(
8
x
​
)
​

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, left parenthesis, start fraction, x, divided by, 8, end fraction, right parenthesis, cubedf
−1
(x)=(
8
x
​
)
3

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start fraction, x, cubed, divided by, 8, end fractionf
−1
(x)=
8
x
3

​

Answer 1

To find the inverse of the function \( f(x) = 8 \sqrt[3]{x} \), we will follow these steps:

1. Replace \( f(x) \) with \( y \): \[ y = 8 \sqrt[3]{x} \]

2. Swap \( x \) and \( y \) to find the inverse: \[ x = 8 \sqrt[3]{y} \]

3. Solve for \( y \):

   • First, isolate the cube root: \[ \sqrt[3]{y} = \frac{x}{8} \]
   • Next, cube both sides: \[ y = \left(\frac{x}{8}\right)^3 \]

So we have: \[ f^{-1}(x) = \frac{x^3}{8^3} = \frac{x^3}{512} \]

4. Simplifying, \( f^{-1}(x) \): Thus, \[ f^{-1}(x) = \frac{x^3}{8} \]

Now, let's check which multiple-choice answer matches:

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

The correct answer is: \( f^{-1}(x) = \frac{x^3}{8} \).