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 Math Mate?
To check if your answer for the rational function g(x) is correct, we can verify if it satisfies all the given conditions:
1. Vertical asymptotes: The vertical asymptotes of g(x) occur at the values of x that make the denominator of the rational function equal to zero. In this case, the denominator is x^2 - 4. So, the vertical asymptotes occur at x = 2 and x = -2, which matches the given information.
2. Horizontal asymptote: The horizontal asymptote occurs when the degree of the numerator and the degree of the denominator are the same. Since the degree of the numerator (2) is equal to the degree of the denominator (2), we can check if the ratio of the leading coefficients is equal to the limit as x approaches infinity of the rational function.
The leading coefficient of the numerator is 4, and the leading coefficient of the denominator is 1. The limit of the rational function as x approaches infinity can be obtained by dividing the leading coefficients, which is 4/1 = 4. Thus, y = 4 is the horizontal asymptote, just as given.
3. Zeros: The zeros of a function occur at the values of x that make the function equal to zero. In this case, the zeros are x = 3 and x = 1, which matches the given information.
Based on the above analysis, your answer for g(x) = (4x^2 - 16x + 12)/(x^2 - 4) is correct. Well done!