A) To find (f+g)(x), we need to add the two functions f(x) and g(x):
(f+g)(x) = f(x) + g(x)
= (3x+1)/(8x-9) + (5x)/(8x-9)
We cannot simplify this any further.
B) The domain of (f+g)(x) will be the set of all real numbers except for the values of x that make the denominator 0. In this case, the denominator is 8x-9. So, the domain of (f+g)(x) is all real numbers except x = 9/8.
C) To find (f/g)(x), we need to divide the function f(x) by g(x):
(f/g)(x) = (f(x))/(g(x))
= (3x+1)/(8x-9) / (5x)/(8x-9)
Dividing by a fraction is the same as multiplying by the reciprocal, so we can rewrite this as:
(f/g)(x) = (3x+1)/(8x-9) * (8x-9)/(5x)
Simplifying,
(f/g)(x) = (3x+1)/(5x)
D) The domain of (f/g)(x) will be the set of all real numbers except for the values of x that make the denominator 0. In this case, the denominator is 5x. So, the domain of (f/g)(x) is all real numbers except x = 0.