Question
Find the inverse of the function \( f(x)=\frac{4}{x-7} \) \( f^{-1}(x)=\square \)
Ask by Summers Stanley. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The inverse function is \( f^{-1}(x) = \frac{4 + 7x}{x} \).
Solution
To find the inverse of the function \( f(x) = \frac{4}{x-7} \), we need to follow these steps:
1. Replace \( f(x) \) with \( y \) in the original function.
2. Swap \( x \) and \( y \) in the equation.
3. Solve the new equation for \( y \) to find the inverse function.
Let's start by replacing \( f(x) \) with \( y \) in the original function:
\[ y = \frac{4}{x-7} \]
Now, we swap \( x \) and \( y \) in the equation:
\[ x = \frac{4}{y-7} \]
Next, we solve the new equation for \( y \) to find the inverse function:
\[ x(y-7) = 4 \]
\[ xy - 7x = 4 \]
\[ xy = 4 + 7x \]
\[ y = \frac{4 + 7x}{x} \]
Therefore, the inverse function \( f^{-1}(x) \) is:
\[ f^{-1}(x) = \frac{4 + 7x}{x} \]
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Simplify this solution The Deep Dive
To find the inverse of the function \( f(x) = \frac{4}{x-7} \), we start by replacing \( f(x) \) with \( y \): \[ y = \frac{4}{x-7} \] Next, we swap \( x \) and \( y \) to solve for the inverse: \[ x = \frac{4}{y-7} \] Now, to solve for \( y \), we multiply both sides by \( y - 7 \): \[ x(y - 7) = 4 \] Expanding this gives us: \[ xy - 7x = 4 \] Now, we isolate \( y \) by adding \( 7x \) to both sides: \[ xy = 4 + 7x \] Next, we divide by \( x \) to solve for \( y \): \[ y = \frac{4 + 7x}{x} \] So the inverse function is: \[ f^{-1}(x) = \frac{7x + 4}{x} \]
