Final answer:
To find R(x), the sum of f(x) and g(x), simplify both functions, f(x) = 2/(x(x - 7)) and g(x) = (7x + 5)/(x(x - 7)), where they share the same denominator. Then add the numerators to get R(x) = (7x + 7)/(x(x - 7)).
Step-by-step explanation:
To find R(x), which is the sum of f(x) and g(x), we need to add the two given functions together. The functions provided are f(x) = \frac{6}{3x^2 - 21x} and g(x) = \frac{7x + 5}{3x^2 - 3x}. First, let's simplify these functions.
f(x) can be simplified by factoring out a 3 from the denominator:
f(x) = \frac{6}{3x(x - 7)} = \frac{2}{x(x - 7)}
Now let's simplify g(x) by factoring out a 3 from the denominator as well:
g(x) = \frac{7x + 5}{3x(x - 1)} = \frac{7x + 5}{3x^2 - 3x}
Having the same denominator for both functions is a critical step before adding them. The denominator for both simplified functions becomes 3x^2 - 21x. Subsequently, we can add the numerators:
R(x) = f(x) + g(x) = \frac{2}{x(x - 7)} + \frac{7x + 5}{x(x - 7)}
Combining the fractions gives us:
R(x) = \frac{2 + (7x + 5)}{x(x - 7)}
Finally, simplifying the numerator:
R(x) = \frac{7x + 7}{x(x - 7)}
Thus, the combined function R(x) is \frac{7x + 7}{x(x - 7)}.