A size 36 shoe in France is size 3.5 in England. A function that converts shoe sizes in France to those in England is g(x)= 3x-94/4. A size 6 shoe in the United States is size 3.5 in England.
A function that converts shoe sizes in England to those in the United States is f(x)=x+5/2.
a. Use composition of functions to find a function that converts shoe sizes in France to those in the United States. Simplify the function.
b. Determine the shoe size in the United States for a size 38 shoe in France.
a. To find a function that converts shoe sizes in France to those in the United States, we need to combine the functions g(x) and f(x) using composition of functions. The composition of functions is denoted by (f o g)(x), where (f o g)(x) = f(g(x)).
In this case, we need to find (f o g)(x) where g(x) is the function that converts shoe sizes in France to those in England and f(x) is the function that converts shoe sizes in England to those in the United States.
Substituting g(x) into f(x), we have:
(f o g)(x) = f(g(x))
Let's substitute g(x) = 3x - 94/4 into f(x):
(f o g)(x) = f(3x - 94/4)
Using the function f(x) = x + 5/2, we substitute 3x - 94/4 into x:
(f o g)(x) = (3x - 94/4) + 5/2
Simplifying further:
(f o g)(x) = 3x - 94/4 + 5/2
= 3x - 47/2 + 5/2
= 3x - 42/2
= 3x - 21
Therefore, the function that converts shoe sizes in France to those in the United States is f(g(x)) = 3x - 21.
b. To determine the shoe size in the United States for a size 38 shoe in France, we need to use the function we found in part (a):
f(g(x)) = 3x - 21
Substituting x = 38 into the function:
f(g(38)) = 3(38) - 21
= 114 - 21
= 93
Therefore, the shoe size in the United States for a size 38 shoe in France is 93.