Question

A dress that is size x in France is equaivalent to size​ d(x) in the United​ States, where ​d(x)equalsxminus32.
​(a). Find a function that will convert U.S. dress size to French dress size.
​(b) Use the inverse function to find the French size of a dress if it is U.S. size 8.
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Part 1
​(a) The inverse of the function is d Superscript negative 1 Baseline left parenthesis x right parenthesis equals nothing
  
.
​(Simplify your​ answer.)

Answer 1

To solve the problem, we will find the inverse function of \( d(x) = x - 32 \).

### Part 1: Finding the inverse function

Given:
\[ d(x) = x - 32 \]

To find the inverse, we will express \( x \) in terms of \( y \):
1. Replace \( d(x) \) with \( y \):
\[ y = x - 32 \]

2. Solve for \( x \):
\[ x = y + 32 \]

Now, we can express the inverse function:
\[ d^{-1}(y) = y + 32 \]

For clarity in notation, we can replace \( y \) with \( x \) to denote the input of the inverse function:
\[ d^{-1}(x) = x + 32 \]

### Final Answer for Part 1
The inverse of the function is:
\[ d^{-1}(x) = x + 32 \]

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### Part 2: Finding the French size for U.S. size 8

Now we will find the French size when the U.S. size is 8 using the inverse function.

\[
d^{-1}(8) = 8 + 32 = 40
\]

### Final Answer for Part 2
The French dress size equivalent to U.S. size 8 is:
40.

In summary:
1. The inverse function is \( d^{-1}(x) = x + 32 \).
2. For U.S. size 8, the French dress size is 40.