Question
Find the inverse of \( \begin{array}{l}f(x)=\sqrt[5]{x-7}+3 \\ f^{-1}(x)=(x-7)^{5}+3 \\ f^{-1}(x)=(x+7)^{5}-3 \\ f^{-1}(x)=(x+3)^{5}-7 \\ f^{-1}(x)=(x-3)^{5}+7\end{array} \)
Ask by Macdonald Dawson. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The inverse function is \( f^{-1}(x) = (x-3)^5 + 7 \).
Solution
Let \( f(x) = \sqrt[5]{x-7}+3 \). To find the inverse function \( f^{-1}(x) \), follow these steps:
1. Replace \( f(x) \) with \( y \):
\[
y = \sqrt[5]{x-7}+3
\]
2. Swap \( x \) and \( y \) to reflect the inverse:
\[
x = \sqrt[5]{y-7}+3
\]
3. Isolate the fifth root by subtracting 3 from both sides:
\[
x - 3 = \sqrt[5]{y-7}
\]
4. Eliminate the fifth root by raising both sides to the power of 5:
\[
(x-3)^5 = y-7
\]
5. Solve for \( y \) by adding 7 to both sides:
\[
y = (x-3)^5 + 7
\]
Thus, the inverse function is:
\[
f^{-1}(x) = (x-3)^5 + 7
\]
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Simplify this solution Beyond the Answer
To find the correct inverse of the function \( f(x)=\sqrt[5]{x-7}+3 \), we start by rearranging to express \( x \) in terms of \( y \). If we set \( y = f(x) \), we have: \[ y = \sqrt[5]{x-7} + 3 \] Subtracting 3 from both sides gives: \[ y - 3 = \sqrt[5]{x - 7} \] Next, we raise both sides to the power of 5: \[ (y - 3)^5 = x - 7 \] Now, adding 7 to both sides results in the expression for \( x \): \[ x = (y - 3)^5 + 7 \] Now we switch \( x \) and \( y \) to find the inverse: \[ f^{-1}(x) = (x - 3)^5 + 7 \] Therefore, the correct answer is: \[ f^{-1}(x) = (x - 3)^5 + 7 \] So the third option \( f^{-1}(x)=(x-3)^{5}+7 \) is the correct inverse function.
