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.
For f(x)=(3x^2)+5and g(x)=7x-2,
a. Verify: g(x + 2) ≠ g(x) + g(2).
b. Find (f – g)(x).
c. Using the resulting function in (b), show that (f – g)(2) = f(2) – g(2).
(The work should be different for each side of the equation.)
d. Is (fg)(0) =(f/g)(0)? Explain.
e. Find (f(x+h)-f(x))/h , h ≠ 0.
a. To find a function that converts shoe sizes in France to those in the United States, we need to compose the functions g(x) and f(x). Let's denote the shoe size in France as x.
First, we apply g(x) to convert from France to England:
g(x) = 3x - 94/4
Then, we apply f(x) to convert from England to the United States:
f(g(x)) = f(3x - 94/4)
= (3x - 94/4) + 5/2
= 3x - 94/4 + 10/4
= 3x - 84/4
= 3x - 21
Therefore, the function that converts shoe sizes in France to those in the United States is h(x) = 3x - 21.
b. To determine the shoe size in the United States for a size 38 shoe in France, we can substitute x = 38 into the function h(x):
h(38) = 3(38) - 21
= 114 - 21
= 93
Therefore, the shoe size in the United States for a size 38 shoe in France is 93.
a. To find a function that converts shoe sizes in France to those in the United States using composition, we first need to compose the given functions in the correct order.
Let's call the function that converts shoe sizes in France to those in England as g(x), and the function that converts shoe sizes in England to those in the United States as f(x).
So, the composition function would be f(g(x)).
To simplify the function, we substitute g(x) into f(x):
f(g(x)) = f(3x - 94/4)
Now, we simplify f(3x - 94/4) using the given function f(x) = (x + 5)/2:
f(3x - 94/4) = ((3x - 94/4) + 5)/2
Simplifying further:
((3x - 94/4) + 5)/2 = (3x - 94/4 + 10/4)/2
= (3x - 84/4)/2
= (3x - 21)/2
Therefore, the function that converts shoe sizes in France to those in the United States is h(x) = (3x - 21)/2.
b. To determine the shoe size in the United States for a size 38 shoe in France, we can substitute x = 38 into the function h(x).
h(38) = (3 * 38 - 21)/2
= (114 - 21)/2
= 93/2
Therefore, the shoe size in the United States for a size 38 shoe in France is 46.5.