Question
For the function f, of, x, equals, start fraction, x, start superscript, one third, end superscript, divided by, 5, end fraction, plus, 7f(x)=
5
x
3
1
+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
5
x
3
1
+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
Answers
GPT-4o mini
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:
**\( f^{-1}(x) = (5(x - 7))^3 \)**.
\[
f(x) = \frac{x^{1/3}}{5} + 7,
\]
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:
**\( f^{-1}(x) = (5(x - 7))^3 \)**.