To find the inverse function that converts U.S. dress sizes to French dress sizes, you can start with the original function provided:
\[ d(x) = x - 32 \]
Part (a): Finding the Inverse Function
To find the inverse, we need to express \( x \) in terms of \( d(x) \):
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Rewrite the function: \[ y = d(x) = x - 32 \]
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Solve for \( x \): \[ y + 32 = x \]
So, the inverse function is: \[ d^{-1}(y) = y + 32 \]
Part (b): Finding the French Size for U.S. Size 12
Now, we can use the inverse function to find the French dress size when the U.S. dress size is 12:
\[ d^{-1}(12) = 12 + 32 = 44 \]
Thus, the French size corresponding to U.S. size 12 is \( \boxed{44} \).