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 20.
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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.)
Part 2
​(b) The French size of a dress if it is U.S. size 20 is
  
enter your response here.

Answer 1

To solve the problem, we'll go through each part step by step.

### Part (a)

We have the function that converts a French dress size \( x \) to a U.S. dress size via the equation:

\[
d(x) = x - 32.
\]

To find the inverse function \( d^{-1}(x) \), we start by letting \( y = d(x) \):

\[
y = x - 32.
\]

Now, we will solve for \( x \) in terms of \( y \):

\[
y + 32 = x.
\]

So, the inverse function that converts a U.S. dress size \( y \) back to a French dress size \( x \) is:

\[
d^{-1}(y) = y + 32.
\]

### Part (b)

Now using the inverse function to find the French size of a dress if it is U.S. size 20, we will substitute \( y = 20 \) into the inverse function:

\[
d^{-1}(20) = 20 + 32 = 52.
\]

Thus, the French size of a dress if it is U.S. size 20 is:

\[
\boxed{52}.
\]

### Summary of Answers
- (a) The inverse function is:

\[
d^{-1}(x) = x + 32.
\]

- (b) The French size of a dress if it is U.S. size 20 is \( 52 \).