To find the inverse of the function \( f(x) = \frac{7x - 4}{x + 3} \), we will follow these steps:
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Replace \( f(x) \) with \( y \): \[ y = \frac{7x - 4}{x + 3} \]
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Swap \( x \) and \( y \) to find the inverse: \[ x = \frac{7y - 4}{y + 3} \]
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Solve for \( y \). First, multiply both sides by \( y + 3 \) to eliminate the fraction: \[ x(y + 3) = 7y - 4 \] \[ xy + 3x = 7y - 4 \]
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Rearrange the equation to isolate \( y \): \[ xy - 7y = -4 - 3x \] \[ y(x - 7) = -4 - 3x \] \[ y = \frac{-4 - 3x}{x - 7} \]
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Thus, we have the inverse function: \[ f^{-1}(x) = \frac{-3x - 4}{x - 7} \]
Now, let's look at the options you provided:
- This function is not one-to-one and therefore has no inverse.
- \( f^{-1}(x) = \frac{-3x - 4}{x - 7} \)
- \( f^{-1}(x) = \frac{-x + 3}{7x - 4} \)
- \( f^{-1}(x) = \frac{3x + 4}{x + 7} \)
The correct answer is: \( f^{-1}(x) = \frac{-3x - 4}{x - 7} \).