To find the inverse function of \( f(x) = -\frac{9}{5}x \), we can follow these steps:
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Replace \( f(x) \) with \( y \): \[ y = -\frac{9}{5}x \]
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Swap \( x \) and \( y \): \[ x = -\frac{9}{5}y \]
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Solve for \( y \): Multiply both sides by \( -\frac{5}{9} \): \[ y = -\frac{5}{9}x \]
Thus, the inverse function is: \[ f^{-1}(x) = -\frac{5}{9}x \]
Now, let's look at the multiple choice options given:
- \( f^{-1}(x) = \frac{5x}{9} \)
- \( f^{-1}(x) = -\frac{9}{5}x \)
- \( f^{-1}(x) = \frac{5}{9}x \)
- \( f^{-1}(x) = -\frac{9}{5}x \)
The correct answer based on our calculation is: \[ f^{-1}(x) = -\frac{5}{9}x \]
None of the given options exactly match this. It seems like there might be an oversight in the options provided, as the correct inverse \( -\frac{5}{9}x \) does not appear in the list.