The function g(x) = (x^2 - 4x - 5)/(x^3 - 6x^2 + 9x) is a rational function. To determine the asymptotes for this function, we need to consider both horizontal and vertical asymptotes.
1. Vertical Asymptotes:
Vertical asymptotes occur when the denominator of a rational function equals zero. In this case, we need to find the values of x that make the denominator, x^3 - 6x^2 + 9x, equal to zero.
To find these values, we can factor the denominator:
x^3 - 6x^2 + 9x = x(x^2 - 6x + 9) = x(x - 3)^2.
Setting each factor equal to zero, we find:
x = 0 (from x = 0)
x - 3 = 0 (from x - 3 = 0)
So, the vertical asymptote(s) for the function g(x) occur at x = 0 and x = 3.
2. Horizontal Asymptotes:
To find the horizontal asymptotes, we need to compare the degrees of the numerator and denominator.
The degree of the numerator is 2 (highest power of x is x^2), and the degree of the denominator is 3 (highest power of x is x^3).
Since the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at y = 0, which is the x-axis.
Therefore, the function g(x) has vertical asymptotes at x = 0 and x = 3, and a horizontal asymptote at y = 0 (the x-axis).
Remember, asymptotes are imaginary lines that the graph of a function approaches but never touches. They can help us understand the behavior of a function as x gets very large or very small.