The table shows men's shoe sizes in the United States and the corresponding European shoe sizes. Let y = f(x)represent the function that gives the men's European shoe size in terms of x, the men's U.S. size.

Men's U.S.
shoe size, Men's European
shoe size
8 41
9 42
10 43
11 45
12 46
13 47

(b) Find f(13).
(c) Find f −1(42), if possible. (If an answer does not exist, enter DNE.)
(d) Find f(f −1(43)).
(e) Find f −1(f(8)).

To find the answers to these questions, we need to understand the relationship between the men's U.S. shoe sizes and the corresponding European shoe sizes.

(a) The given function, y = f(x), represents the relationship between the men's European shoe size and the U.S. shoe size. So, to find f(13), we need to find the corresponding European shoe size for a U.S. shoe size of 13.

From the table, we see that for a U.S. size of 13, the corresponding European size is 47. Therefore, f(13) = 47.

(b) To find f −1(42), we need to find the U.S. shoe size that corresponds to a European shoe size of 42. However, in this case, there is no value of x in the table that corresponds to a European shoe size of 42. Therefore, f −1(42) = DNE (Does Not Exist).

(c) To find f(f −1(43)), we need to find the corresponding European shoe size for a U.S. shoe size that corresponds to a European shoe size of 43. Basically, we need to find f(x), where x = f −1(43).

From the table, we see that for a U.S. size of 10, the corresponding European size is 43. Therefore, f −1(43) = 10. Now, we can substitute x = 10 into the function y = f(x) to find f(f −1(43)).

f(f −1(43)) = f(10) = 43.

Therefore, f(f −1(43)) = 43.

(d) To find f −1(f(8)), we need to find the U.S. shoe size that corresponds to a European shoe size that corresponds to a U.S. shoe size of 8. Basically, we need to find f −1(x), where x = f(8).

From the table, we see that for a U.S. size of 8, the corresponding European size is 41. Therefore, f(8) = 41. Now, we can substitute x = 41 into the inverse function y = f −1(x) to find f −1(f(8)).

f −1(f(8)) = f −1(41).

Using the table, we can see that there is no value of x that corresponds to a European shoe size of 41. Therefore, f −1(f(8)) = DNE (Does Not Exist).

(e) Similarly, to find f −1(f(8)), we need to find the U.S. shoe size that corresponds to a European shoe size that corresponds to a U.S. shoe size of 8. However, based on the table, there is no U.S. shoe size 8 listed. Therefore, f(8) is undefined, and we cannot find its inverse, f −1(f(8)).