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 analyze the given table:
Men's U.S. shoe size, Men's European shoe size
8 41
9 42
10 43
11 45
12 46
13 47
(a) The function, y = f(x), gives the men's European shoe size in terms of the men's U.S. shoe size.
(b) To find f(13), we need to find the European shoe size when the U.S. shoe size is 13. From the table, we can see that f(13) = 47.
Answer: f(13) = 47
(c) To find f^(-1)(42), we need to find the U.S. shoe size when the European shoe size is 42. Looking at the table, we can see that there is no direct value for 42 in the European shoe size column. Therefore, the inverse function f^(-1)(42) does not exist.
Answer: f^(-1)(42) = DNE (Does Not Exist)
(d) To find f(f^(-1)(43)), we need to calculate the European shoe size when the U.S. shoe size is f^(-1)(43). First, we need to find f^(-1)(43). From the table, we can see that f^(-1)(43) = 10. Now we can use f(10) to find the corresponding European shoe size, which is 43.
Answer: f(f^(-1)(43)) = f(10) = 43
(e) To find f^(-1)(f(8)), we need to calculate the U.S. shoe size when the European shoe size is f(8). First, we need to find f(8) using the table, which is 41. Now we can use f^(-1)(41) to find the corresponding U.S. shoe size. However, looking at the table, there is no direct value for 41 in the European shoe size column. Therefore, the inverse function f^(-1)(41) does not exist.
Answer: f^(-1)(f(8)) = DNE (Does Not Exist)
To find the answers to these questions, we need to understand and interpret the given table.
In the table, the first column represents the men's U.S. shoe sizes, and the second column represents the corresponding men's European shoe sizes. The function y = f(x) represents the relationship between the men's U.S. shoe size (x) and the men's European shoe size (y).
(b) To find f(13), we need to determine the men's European shoe size corresponding to the U.S. size 13. From the table, we can see that the value for the European shoe size when the U.S. size is 13 is 47. Therefore, f(13) = 47.
(c) To find f −1(42), we need to find the men's U.S. shoe size corresponding to the European size 42. To do this, we need to look for the value 42 in the second column of the table. However, in this case, there is no exact match for 42 in the table, so the inverse function value f −1(42) does not exist (DNE).
(d) To find f(f −1(43)), we need to find the value of f −1(43) first and then substitute it back into the function f(x). Since f −1(43) represents the men's U.S. shoe size corresponding to the European size 43, we need to find the value in the first column of the table that matches 43 in the second column. From the table, we can see that the U.S. shoe size when the European size is 43 is 10. Therefore, f −1(43) = 10. Plugging this value back into the function, we have f(f −1(43)) = f(10). From the table, we can see that the European shoe size when the U.S. size is 10 is 43. Therefore, f(f −1(43)) = f(10) = 43.
(e) To find f −1(f(8)), we need to find the inverse function value of f(8). Since f(8) represents the men's European shoe size corresponding to the U.S. size 8, we need to find the value in the second column of the table that matches 8 in the first column. From the table, we can see that the European shoe size when the U.S. size is 8 is 41. Therefore, f(8) = 41. Plugging this value into the inverse function, we have f −1(f(8)) = f −1(41). However, there is no exact match for 41 in the second column of the table, so the inverse function value f −1(41) does not exist (DNE).
In summary:
(b) f(13) = 47
(c) f −1(42) = DNE (does not exist)
(d) f(f −1(43)) = f(10) = 43
(e) f −1(f(8)) = f −1(41) = DNE (does not exist)
I don't understand what problem you are having with these, the answers are just staring at you
e.g. f(13) = 47
etc
and remember that f(f^-1 (x) ) = f^-1 (f(x)) = x