Question

For the function f, of, x, equals, 8, cube root of, x, end cube rootf(x)=8
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:
**\( f^{-1}(x) = \frac{x^3}{8} \)**.