Question

For the function f, of, x, equals, start fraction, cube root of, x, end cube root, minus, 7, divided by, 5, end fractionf(x)=
5
3

x
​
−7
​
, find f, to the power minus 1 , left parenthesis, x, right parenthesisf
−1
(x).
Answer
Attempt 1 out of 2
Multiple Choice Answers
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 5, x, cubed, plus, 7f
−1
(x)=5x
3
+7
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 5, left parenthesis, x, plus, 7, right parenthesis, cubedf
−1
(x)=5(x+7)
3

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, left parenthesis, 5, left parenthesis, x, plus, 7, right parenthesis, right parenthesis, cubedf
−1
(x)=(5(x+7))
3

f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, left parenthesis, 5, x, plus, 7, right parenthesis, cubedf
−1
(x)=(5x+7)
3

Answer 1

To find the inverse function \( f^{-1}(x) \) for the given function \( f(x) = \frac{\sqrt[3]{x} - 7}{5} \), we need to follow these steps:

1. **Replace \( f(x) \) with \( y \)**:
\[
y = \frac{\sqrt[3]{x} - 7}{5}
\]

2. **Swap \( x \) and \( y \)**:
\[
x = \frac{\sqrt[3]{y} - 7}{5}
\]

3. **Solve for \( y \)**:
- Multiply both sides by 5:
\[
5x = \sqrt[3]{y} - 7
\]
- Add 7 to both sides:
\[
5x + 7 = \sqrt[3]{y}
\]
- Cube both sides:
\[
(5x + 7)^3 = y
\]

4. **Write the inverse function**:
\[
f^{-1}(x) = (5x + 7)^3
\]

Now, let's check the available options:

1. **Option 1**: \( f^{-1}(x) = 5x^3 + 7 \)
2. **Option 2**: \( f^{-1}(x) = 5(x + 7)^3 \)
3. **Option 3**: \( f^{-1}(x) = (5(x + 7))^3 \)
4. **Option 4**: \( f^{-1}(x) = (5x + 7)^3 \)

The correct option is:
\[
f^{-1}(x) = (5x + 7)^3
\]
which corresponds to **Option 4**.