which of the following is an asymptote of f(x)= 5/x+6
I hope one of the choices was y=5
To find the asymptote(s) of the function f(x) = 5/(x + 6), we need to determine the vertical and horizontal asymptotes.
Vertical asymptotes occur when the denominator of the function approaches zero. In this case, the denominator is (x + 6). To find the value that makes the denominator zero, we set (x + 6) = 0 and solve for x:
x + 6 = 0
x = -6
So, there is a vertical asymptote at x = -6.
Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. In general, to find the horizontal asymptote, we compare the degrees of the numerator and the denominator. In this case, the degree of the numerator is 0 and the degree of the denominator is 1. Since the degree of the numerator is lower than the degree of the denominator, the horizontal asymptote can be found by taking the ratio of the leading coefficients of the numerator and the denominator.
In the given function, the leading coefficient of the numerator is 5 and the leading coefficient of the denominator is 1. So, the horizontal asymptote is y = 5/1, which simplifies to y = 5.
Therefore, the asymptotes of the function f(x) = 5/(x + 6) are:
- Vertical asymptote: x = -6
- Horizontal asymptote: y = 5