Answer:
The equations of the asymptotes of the graph of f(x) = (3/(x-2))+9 are:
-Vertical asymptote: x = 2
-Horizontal asymptote: y = 0.
Explanation:
To find the equations of the asymptotes of the graph of the function f(x) = (3/(x-2))+9, we first need to determine the vertical and horizontal asymptotes.
Vertical Asymptote:
A vertical asymptote occurs when the denominator of a rational function equals zero, since division by zero is undefined. In this case, the denominator of f(x) is (x-2), which equals zero when x = 2. Therefore, there is a vertical asymptote at x = 2.
Horizontal Asymptote:
To find the horizontal asymptote, we need to look at the behavior of the function as x approaches positive or negative infinity. Since the highest power of x in the denominator is 1 and the highest power of x in the numerator is also 1, the degree of the numerator and denominator are the same. Therefore, we can find the horizontal asymptote by dividing the leading coefficient of the numerator by the leading coefficient of the denominator. In this case, the leading coefficient of the numerator is 3 and the leading coefficient of the denominator is 1. So, the horizontal asymptote is y = 0.
Therefore, the equations of the asymptotes of the graph of f(x) = (3/(x-2))+9 are:
-Vertical asymptote: x = 2
-Horizontal asymptote: y = 0.