Final answer:
The rational function we are looking for is f(x) = 2x / (x² - 9), which renders option (a) as the correct answer with an adjusted constant.
Step-by-step explanation:
The question requires us to determine the equation of a rational function that has specific characteristics. We know the function passes through the points (0,0) and (4,8/7), the x-axis is the horizontal asymptote, and there are vertical asymptotes at x=3 and x=-3. These criteria suggest a function with a numerator linear in x (so it can pass through the origin) and a denominator that is zero at x=3 and x=-3.
From this information, we can conclude that the denominator must be a factor of (x-3) and (x+3), which gives us a denominator of (x2 - 9). Since the function passes through (4, 8/7), we need to determine the constant multiplier for the x term in the numerator that will satisfy this requirement.
To satisfy the point (4, 8/7), our function must be f(x) = k * x / (x2 - 9) where k is found by substituting x=4 to get 8/7 = k * 4 / (42 - 9). Solving for k, we find k=2. Thus the correct equation for the rational function is:
f(x) = 2x / (x2 - 9)
Therefore, the correct option is (a) f(x)=4x/x2−9 adjusted for the correct constant k.