Lang Reid
02/28/2023 · Junior High School
Let \( f(x)=7+\sqrt{6 x-5} \). Find \( f^{-1}(x) \) \( f^{-1}(x)=\square \)
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Tutor-Verified Answer
Step-by-step Solution
To find the inverse function \( f^{-1}(x) \) of the function \( f(x) = 7 + \sqrt{6x - 5} \), we will follow these steps:
1. **Set \( f(x) \) equal to \( y \)**:
\[
y = 7 + \sqrt{6x - 5}
\]
2. **Solve for \( x \) in terms of \( y \)**:
- First, isolate the square root:
\[
y - 7 = \sqrt{6x - 5}
\]
- Next, square both sides to eliminate the square root:
\[
(y - 7)^2 = 6x - 5
\]
- Rearranging gives:
\[
6x = (y - 7)^2 + 5
\]
- Now, solve for \( x \):
\[
x = \frac{(y - 7)^2 + 5}{6}
\]
3. **Replace \( y \) with \( x \) to express the inverse function**:
\[
f^{-1}(x) = \frac{(x - 7)^2 + 5}{6}
\]
Thus, the inverse function is:
\[
\boxed{f^{-1}(x) = \frac{(x - 7)^2 + 5}{6}}
\]
Quick Answer
\( f^{-1}(x) = \frac{(x - 7)^2 + 5}{6} \)
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