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).
Given:
f(x) = x+2/x-5
g(x) = (6x-12)/(x^2 + 25)
To add two functions, we add the corresponding terms together.
(f+g)(x) = f(x) + g(x)
First, let's simplify the expression for f(x):
f(x) = x+2/x-5
= (x*(x-5) + 2)/(x-5)
= (x^2 - 5x + 2)/(x-5)
Now, let's simplify the expression for g(x):
g(x) = (6x-12)/(x^2 + 25)
Next, let's add the two functions together:
(f+g)(x) = (x^2 - 5x + 2)/(x-5) + (6x-12)/(x^2 + 25)
To add fractions, we need to find a common denominator. The common denominator for (x-5) and (x^2 + 25) is (x-5)*(x^2 + 25).
(f+g)(x) = ((x^2 - 5x + 2)*(x^2 + 25) + (6x-12)*(x-5))/((x-5)*(x^2 + 25))
Now, let's expand and simplify the numerator:
((x^2 - 5x + 2)*(x^2 + 25) + (6x-12)*(x-5)) = (x^4 - 3x^3 - 23x^2 + 75x + 50)/((x-5)*(x^2 + 25))
Therefore, the expression for (f+g)(x) is:
(f+g)(x) = (x^4 - 3x^3 - 23x^2 + 75x + 50)/((x-5)*(x^2 + 25))