For the function f, of, x, equals, start fraction, x, start superscript, one third, end superscript, divided by, 5, end fraction, plus, 7f(x)=

Question

5
x
3
1
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+7, 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, 5, x, minus, 7, right parenthesis, cubedf
−1
(x)=(5x−7)
3

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

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

Answer 1

Given the function

\[ f(x) = \frac{x^{1/3}}{5} + 7, \]

we want to find the inverse function \( f^{-1}(x) \).

To find the inverse, we first rewrite the equation as:

\[ y = \frac{x^{1/3}}{5} + 7. \]

To find \( f^{-1}(x) \), we will follow these steps:

1. Replace \( f(x) \) with \( y \): \[ y = \frac{x^{1/3}}{5} + 7. \]

2. Solve for \( x \) in terms of \( y \): Start by isolating the term with \( x \):

\[ y - 7 = \frac{x^{1/3}}{5}. \]

Next, multiply both sides by 5:

\[ 5(y - 7) = x^{1/3}. \]

3. Now, cube both sides to solve for \( x \):

\[ x = \left(5(y - 7)\right)^3. \]

4. To express the inverse function, replace \( y \) back with \( x \):

\[ f^{-1}(x) = \left(5(x - 7)\right)^3. \]

Now let's look at the multiple-choice answers provided:

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

The correct answer is:

\[ f^{-1}(x) = (5(x - 7))^3. \]

So, the answer is:

\( f^{-1}(x) = (5(x - 7))^3 \).