To find the inverse of the function \( f(x) = \frac{x - 5}{x - 7} \), we need to follow these steps:
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Replace \( f(x) \) with \( y \): \[ y = \frac{x - 5}{x - 7} \]
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Swap \( x \) and \( y \) to find the inverse: \[ x = \frac{y - 5}{y - 7} \]
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Solve for \( y \):
- Multiply both sides by \( (y - 7) \): \[ x(y - 7) = y - 5 \]
- Distributing \( x \) gives: \[ xy - 7x = y - 5 \]
- Rearranging terms to isolate \( y \): \[ xy - y = 7x - 5 \]
- Factor \( y \) out: \[ y(x - 1) = 7x - 5 \]
- Now, dividing both sides by \( (x - 1) \) gives: \[ y = \frac{7x - 5}{x - 1} \]
Thus, the inverse function is: \[ f^{-1}(x) = \frac{7x - 5}{x - 1} \]
Next, we also note the restriction for the inverse. Since \( y = \frac{7x - 5}{x - 1} \) will not be defined when \( x = 1 \).
Looking at the answer choices, we see:
D. \( f^{-1}(x) = \frac{7x - 5}{x - 1} \), for \( x \neq 1 \)
is the correct answer.
Thus, the answer is: D. \( f^{-1}(x) = \frac{7x - 5}{x - 1} \), for \( x \neq 1 \).