f(x)=x+2/x-5, g(x) = (6X-12)/(x^2 + 25) what is (f+g)(x)?
To find (f+g)(x), we need to add the functions f(x) and g(x). Let's first rewrite the functions:
f(x) = (x+2)/(x-5)
g(x) = (6x-12)/(x^2 + 25)
To find (f+g)(x), we add the two functions together:
(f+g)(x) = f(x) + g(x) = (x+2)/(x-5) + (6x-12)/(x^2 + 25)
Since the denominators are different, we need to find a common denominator. In this case, the common denominator is (x-5)(x^2 + 25). We can rewrite the fractions with the common denominator:
(f+g)(x) = [(x+2)(x^2 + 25) + (6x-12)(x-5)] / [(x-5)(x^2 + 25)]
Now, let's simplify the numerator:
(f+g)(x) = (x(x^2 + 25) + 2(x^2 + 25) + 6x(x-5) - 12(x-5)) / [(x-5)(x^2 + 25)]
Expanding the terms:
(f+g)(x) = (x^3 + 25x + 2x^2 + 50 + 6x^2 - 30x - 12x + 60) / [(x-5)(x^2 + 25)]
Combining like terms:
(f+g)(x) = (x^3 + 8x^2 - 17x + 110) / [(x-5)(x^2 + 25)]
Therefore, the expression for (f+g)(x) is:
(f+g)(x) = (x^3 + 8x^2 - 17x + 110) / [(x-5)(x^2 + 25)]