For the function f, of, x, equals, start fraction, start root, start index, 7, end index, x, end root, divided by, 3, end fractionf(x)=
3
7
x
, 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, 3, x, right parenthesis, to the power 7f
−1
(x)=(3x)
7
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, start root, start index, 7, end index, left parenthesis, 3, x, right parenthesis, end rootf
−1
(x)=
7
(3x)
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 3, start root, start index, 7, end index, x, end rootf
−1
(x)=3
7
x
f, to the power minus 1 , left parenthesis, x, right parenthesis, equals, 3, x, to the power 7f
−1
(x)=3x
7
1 answer
Given the function
\[
f(x) = \frac{\sqrt[7]{x}}{3}
\]
we want to find the inverse function, \( f^{-1}(x) \).
Starting from the equation of the function:
\[
f(x) = \frac{\sqrt[7]{x}}{3}
\]
To find the inverse, we set \( y = f(x) \):
\[
y = \frac{\sqrt[7]{x}}{3}
\]
Express \( x \) in terms of \( y \):
\[
y \cdot 3 = \sqrt[7]{x}
\]
\[
3y = \sqrt[7]{x}
\]
Now, raise both sides to the power of 7 to eliminate the root:
\[
(3y)^7 = x
\]
Thus, we find the inverse:
\[
f^{-1}(x) = (3x)^7
\]
Now we can check the multiple choice answers provided:
1 answer