Question
Find a formula for the inverse of the function. \[ f(x)=e^{8 x-7} \] \( f^{-1}(x)=\square \)
Ask by Lowe Bartlett. in the United States
Jan 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The inverse function is \( f^{-1}(x) = \frac{\ln(x) + 7}{8} \).
Solution
To find the inverse of the function \( f(x) = e^{8x - 7} \), follow these steps:
1. **Start with the equation:**
\[
y = e^{8x - 7}
\]
2. **Take the natural logarithm of both sides to solve for \( x \):**
\[
\ln(y) = 8x - 7
\]
3. **Isolate \( x \):**
\[
8x = \ln(y) + 7
\]
4. **Divide by 8:**
\[
x = \frac{\ln(y) + 7}{8}
\]
Thus, the inverse function \( f^{-1}(x) \) is:
\[
f^{-1}(x) = \frac{\ln(x) + 7}{8}
\]
**Answer:**
\[ f^{-1}(x) = \frac{\ln(x)\,+\,7}{\,8\,} \]
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Simplify this solution Extra Insights
To find the inverse of the function \( f(x) = e^{8x - 7} \), we start by replacing \( f(x) \) with \( y \): \[ y = e^{8x - 7} \] Next, we solve for \( x \) in terms of \( y \). First, take the natural logarithm of both sides: \[ \ln(y) = 8x - 7 \] Next, isolate \( x \): \[ 8x = \ln(y) + 7 \] \[ x = \frac{\ln(y) + 7}{8} \] Now, we can express the inverse function by swapping \( x \) and \( y \): \[ f^{-1}(x) = \frac{\ln(x) + 7}{8} \] So, the final answer for the inverse function is: \[ f^{-1}(x) = \frac{\ln(x) + 7}{8} \]
