Paul Harper
06/04/2024 · Middle School
Given: \( f(x)=\frac{5 x-1}{1 x-4} \) Find the inverse function, \( f^{-1}(x) \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find the inverse function \( f^{-1}(x) \) of the function \( f(x) = \frac{5x - 1}{x - 4} \), we will follow these steps:
1. Replace \( f(x) \) with \( y \):
\[
y = \frac{5x - 1}{x - 4}
\]
2. Swap \( x \) and \( y \):
\[
x = \frac{5y - 1}{y - 4}
\]
3. Solve for \( y \):
- First, multiply both sides by \( y - 4 \) to eliminate the fraction:
\[
x(y - 4) = 5y - 1
\]
- Distributing \( x \):
\[
xy - 4x = 5y - 1
\]
- Rearranging the equation to isolate terms involving \( y \):
\[
xy - 5y = 4x - 1
\]
- Factor out \( y \) on the left side:
\[
y(x - 5) = 4x - 1
\]
- Finally, solve for \( y \):
\[
y = \frac{4x - 1}{x - 5}
\]
4. Thus, the inverse function is:
\[
f^{-1}(x) = \frac{4x - 1}{x - 5}
\]
To summarize, the inverse function is:
\[
\boxed{f^{-1}(x) = \frac{4x - 1}{x - 5}}
\]
Quick Answer
The inverse function \( f^{-1}(x) \) is \( \frac{4x - 1}{x - 5} \).
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