Final answer:
The student is asked to find algebraic combinations and a function evaluation for f(x) and g(x), and determine domains. Operations are carried out to add, subtract, evaluate, and divide the functions with careful consideration of domain restrictions.
Step-by-step explanation:
The student is asking to find the functions f+g, f-g, g(1), and (f)/(g), and to determine the domain for each function given that f(x) = (8x)/(x-8) and g(x) = (3)/(x+9). We will perform these operations step-by-step.
- To find f+g, we add f(x) and g(x): f(x) + g(x) = (8x)/(x-8) + (3)/(x+9).
- To find f-g, we subtract g(x) from f(x): f(x) - g(x) = (8x)/(x-8) - (3)/(x+9).
- To find g(1), we evaluate g(x) at x=1: g(1) = (3)/(1+9) = 3/10.
- To find (f)/(g), we divide f(x) by g(x): (f(x))/(g(x)) = [(8x)/(x-8)]/[(3)/(x+9)].
- The domain of each function will exclude the values where the denominators are zero, so for f+g and f-g, the domain excludes x = 8 and x = -9, whereas for (f)/(g), the domain also excludes x = 8 and x = -9.