Question
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
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
Answers
GPT-4o mini
Given the function
\[
f(x) = \frac{\sqrt[7]{x}}{3}
\]
we want to find the inverse function, \( f^{-1}(x) \).
1. **Starting from the equation of the function**:
\[
f(x) = \frac{\sqrt[7]{x}}{3}
\]
2. **To find the inverse**, we set \( y = f(x) \):
\[
y = \frac{\sqrt[7]{x}}{3}
\]
3. **Express \( x \) in terms of \( y \)**:
\[
y \cdot 3 = \sqrt[7]{x}
\]
\[
3y = \sqrt[7]{x}
\]
4. **Now, raise both sides to the power of 7** to eliminate the root:
\[
(3y)^7 = x
\]
5. **Thus, we find the inverse**:
\[
f^{-1}(x) = (3x)^7
\]
Now we can check the multiple choice answers provided:
- \( f^{-1}(x) = (3x)^7 \) ✔️
So, the correct answer is:
\[
f^{-1}(x) = (3x)^7
\]
\[
f(x) = \frac{\sqrt[7]{x}}{3}
\]
we want to find the inverse function, \( f^{-1}(x) \).
1. **Starting from the equation of the function**:
\[
f(x) = \frac{\sqrt[7]{x}}{3}
\]
2. **To find the inverse**, we set \( y = f(x) \):
\[
y = \frac{\sqrt[7]{x}}{3}
\]
3. **Express \( x \) in terms of \( y \)**:
\[
y \cdot 3 = \sqrt[7]{x}
\]
\[
3y = \sqrt[7]{x}
\]
4. **Now, raise both sides to the power of 7** to eliminate the root:
\[
(3y)^7 = x
\]
5. **Thus, we find the inverse**:
\[
f^{-1}(x) = (3x)^7
\]
Now we can check the multiple choice answers provided:
- \( f^{-1}(x) = (3x)^7 \) ✔️
So, the correct answer is:
\[
f^{-1}(x) = (3x)^7
\]