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
Attempt 1 out of 2
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

To find the inverse function \( f^{-1}(x) \) for the given function

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

we want to express \( x \) in terms of \( y \) where \( y = f(x) \).

Let's start by setting \( y = f(x) \):

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

Now, we solve for \( x \):

1. Subtract 7 from both sides:

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

2. Multiply both sides by 5:

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

3. Now, cube both sides to eliminate the cube root:

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

This means we can express \( f^{-1}(x) \) as:

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

Thus, the correct multiple choice answer is:

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