Final answer:
To find (f+g)(x), we add f(x) and g(x) together by finding a common denominator and combining the numerators, which results in (f+g)(x) = (5x^2 - 33x + 85) / (x^2 - 5x - 50).
Step-by-step explanation:
To find the sum of the functions f(x) and g(x), denoted as (f+g)(x), you combine the functions by adding them together. Given:
f(x) = (x+7)/(x-10)
g(x) = (4x-5)/(x+5)
To add these two functions, find a common denominator and combine the numerators:
(f+g)(x) = f(x) + g(x)
= [(x+7)(x+5) + (4x-5)(x-10)] / [(x-10)(x+5)]
Now, expand the numerators and combine like terms:
[(x^2+5x+7x+35) + (4x^2-40x-5x+50)] / [(x-10)(x+5)]
= (5x^2 - 33x + 85) / (x^2 - 5x - 50)
Thus, (f+g)(x) = (5x^2 - 33x + 85) / (x^2 - 5x - 50).