f(x)=(7x+28)/(x-3) and g(x)=(3x+28)/(x-7)
Find: (fog)(x)
(gof)(x)
(fog)(-1)
f(g) = (7g+28)/(g-3)
= (7(3x+28)/(x-7) + 28)/(3x+28)/(x-7) - 3)
= x
g(f) = (3f+28)/(f-7)
= (3(7x+28)/(x-3) + 28) / ((7x+28)/(x-3) - 7)
= x
How odd! However, if you solve for f-1(x) you get g(x) and vice-versa.
f(g(-1)) = f(-1) = -1
To find (fog)(x), we need to substitute g(x) into f(x) by replacing every occurrence of x in f(x) with g(x). Similarly, to find (gof)(x), we need to substitute f(x) into g(x) by replacing every occurrence of x in g(x) with f(x).
1. (fog)(x):
Substituting g(x) into f(x), we have:
f(x) = (7x + 28)/(x - 3)
g(x) = (3x + 28)/(x - 7)
Substituting g(x) into f(x):
(fog)(x) = f(g(x)) = f((3x + 28)/(x - 7))
2. (gof)(x):
Substituting f(x) into g(x), we have:
f(x) = (7x + 28)/(x - 3)
g(x) = (3x + 28)/(x - 7)
Substituting f(x) into g(x):
(gof)(x) = g(f(x)) = g((7x + 28)/(x - 3))
3. (fog)(-1):
To find (fog)(-1), we substitute -1 into the expression (fog)(x) obtained in step 1:
(fog)(-1) = f(g(-1)) = f((3(-1) + 28)/(-1 - 7))
These expressions cannot be simplified further since the functions f(x) and g(x) are given in their simplified forms. You can calculate the final values by substituting the appropriate values for x into the expressions.