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 1

To find the inverse function \( f^{-1}(x) \) of the function \( f(x) = 8 \sqrt[3]{x} \), follow these steps:

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

2. **Swap \( x \) and \( y \):**
\[
x = 8 \sqrt[3]{y}
\]

3. **Solve for \( y \):**
- First, divide both sides by 8:
\[
\frac{x}{8} = \sqrt[3]{y}
\]
- Next, cube both sides to eliminate the cube root:
\[
\left( \frac{x}{8} \right)^3 = y
\]

4. **Express \( y \) in terms of \( x \):**
\[
y = \frac{x^3}{8^3}
\]
Since \( 8^3 = 512 \):
\[
y = \frac{x^3}{512}
\]

5. **Rewrite the equation to express the inverse function:**
\[
f^{-1}(x) = \frac{x^3}{512}
\]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{x^3}{512}
\]

Answer 2

it says thats incorrect

Answer 3

Let’s go through the steps again to find the inverse function more carefully.

Given the function:
\[
f(x) = 8 \sqrt[3]{x}
\]

### Step 1: Set the function equal to \( y \)
\[
y = 8 \sqrt[3]{x}
\]

### Step 2: Swap \( x \) and \( y \)
\[
x = 8 \sqrt[3]{y}
\]

### Step 3: Solve for \( y \)
1. Divide both sides by 8:
\[
\frac{x}{8} = \sqrt[3]{y}
\]

2. Cube both sides to eliminate the cube root:
\[
\left( \frac{x}{8} \right)^3 = y
\]

### Step 4: Simplify the expression
\[
y = \frac{x^3}{8^3}
\]
Since \( 8^3 = 512 \), we have:
\[
y = \frac{x^3}{512}
\]

### Step 5: Write the inverse function
So, the inverse function is:
\[
f^{-1}(x) = \frac{x^3}{512}
\]

### Conclusion
The result \( f^{-1}(x) = \frac{x^3}{512} \) should indeed be correct based on the calculations performed. If there is an error being pointed out (such as in a specific context), please clarify or provide what the expected output is so I can address it accordingly.