Question

Find the inverse function of the function f, of, x, equals, 4, cube root of, x, end cube rootf(x)=4
3

x
​
.
Answer
Multiple Choice Answers
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start fraction, x, cubed, divided by, 64, end fractionf
−1
(x)=
64
x
3

​

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 64, x, cubedf
−1
(x)=64x
3

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start fraction, x, cubed, divided by, 4, end fractionf
−1
(x)=
4
x
3

​

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 4, x, cubedf
−1
(x)=4x
3

Answer 1

To find the inverse function of \( f(x) = 4 \sqrt[3]{x} \), we will follow these steps:

1. **Replace \( f(x) \) with \( y \)**:
\[
y = 4 \sqrt[3]{x}
\]

2. **Solve for \( x \)** in terms of \( y \)**:
\[
\frac{y}{4} = \sqrt[3]{x}
\]

3. **Cube both sides** to eliminate the cube root:
\[
\left( \frac{y}{4} \right)^3 = x
\]

4. **Simplify the expression**:
\[
x = \frac{y^3}{4^3} = \frac{y^3}{64}
\]

5. **Replace \( y \) with \( x \)** to denote the inverse function:
\[
f^{-1}(x) = \frac{x^3}{64}
\]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{x^3}{64}
\]

### Answer
The correct choice is:
\[
f^{-1}(x) = \frac{x^3}{64}
\]