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, it is correct, check my response for more information.
http://www.jiskha.com/display.cgi?id=1283787460
To determine if your rational function g(x)=(4x^2-16x+12)/(x^2-4) is correct, we can check if it satisfies all the given conditions.
1. Vertical Asymptotes: The vertical asymptotes occur where the denominator of the rational function is equal to zero, so let's check:
For x = 2:
(x^2 - 4) evaluates to (2^2 - 4) = 4 - 4 = 0
For x = -2:
(x^2 - 4) evaluates to (-2^2 - 4) = 4 - 4 = 0
Since g(x) has (x - 2) and (x + 2) in the denominator, x = 2 and x = -2 are vertical asymptotes. So, this condition is satisfied.
2. Horizontal Asymptote: The horizontal asymptote of a rational function occurs when the degree of the numerator is less than or equal to the degree of the denominator. Let's check:
The highest power of x in the numerator g(x)=(4x^2-16x+12) is 2, and the highest power of x in the denominator (x^2-4) is also 2. Therefore, the degree of the numerator and denominator are the same, and in this case, the horizontal asymptote is given by the ratio of the leading coefficients of the numerator and denominator.
The ratio of the leading coefficients is 4/1 = 4. Hence, y = 4 is the equation of the horizontal asymptote. So, this condition is satisfied.
3. Zeros: The zeros of a rational function occur when the numerator is equal to zero. Let's check:
Setting the numerator g(x) = 0:
4x^2 - 16x + 12 = 0
We can factor the quadratic expression:
4(x^2 - 4x + 3) = 0
4(x - 1)(x - 3) = 0
So, the zeros of g(x) are x = 1 and x = 3. This condition is satisfied as well.
Therefore, based on the given conditions, and considering that the calculated rational function g(x) = (4x^2-16x+12)/(x^2-4) satisfies all those conditions, your answer is correct.