A rational function g has the lines x=2 and x=-2 as vertical asymptotes, the line y=4 as a horizontal asymptote, and numbers 3 and 1 as zeros. Find a formula for g(x).
So I got g(x)=(4x^2-16x+12)/(x^2-4)
Is this correct
Yes, you are correct
Yes, the formula you provided for g(x) is correct. The function g(x) can be written as:
g(x) = (4x^2 - 16x + 12) / (x^2 - 4)
This satisfies the given conditions:
1. The lines x = 2 and x = -2 are vertical asymptotes of the function g(x).
2. The line y = 4 is a horizontal asymptote of the function g(x).
3. The zeros of g(x) are x = 3 and x = 1.
To verify if your answer is correct, we can check if the given function meets all the given conditions.
First, let's check the vertical asymptotes. The function g(x) has vertical asymptotes at x = 2 and x = -2. We can determine this by checking the denominator of the rational function. In your case, the denominator is (x^2 - 4). When the denominator is equal to zero, the function is undefined. Solving (x^2 - 4) = 0, we find that x = 2 and x = -2. Therefore, the vertical asymptotes are correctly specified.
Next, let's check the horizontal asymptote. The function g(x) has a horizontal asymptote at y = 4. This happens when the degree of the numerator is less than or equal to the degree of the denominator. In your case, the degrees are both 2, so we need to compare the leading coefficients. The leading coefficient of the numerator is 4, and the leading coefficient of the denominator is 1. Since 4/1 = 4, the function satisfies the condition of having y = 4 as the horizontal asymptote.
Finally, let's check the zeros. The function g(x) has zeros at 3 and 1. This means that g(3) = 0 and g(1) = 0. We can check this by evaluating the function at these values.
When x = 3:
g(3) = (4(3)^2 - 16(3) + 12)/((3)^2 - 4) = (4(9) - 48 + 12)/(9 - 4) = (36 - 48 + 12)/(5) = 0/5 = 0
When x = 1:
g(1) = (4(1)^2 - 16(1) + 12)/((1)^2 - 4) = (4(1) - 16 + 12)/(1 - 4) = (4 - 16 + 12)/(-3) = 0/(-3) = 0
Both g(3) and g(1) are equal to zero, so the function satisfies the condition of having zeros at 3 and 1.
Based on the verification of the vertical asymptotes, horizontal asymptote, and zeros, your answer g(x) = (4x^2 - 16x + 12)/(x^2 - 4) is indeed correct.