Final answer:
The function f(x) = 4x / (x - 16) has a vertical asymptote at x = 16 and a horizontal asymptote at y = 4. The end behavior of the function is that as x approaches infinity or negative infinity, f(x) approaches the horizontal asymptote y = 4.
Step-by-step explanation:
To identify the asymptotes and state the end behavior of the function f(x) = 4x / (x - 16), we need first to recognize that this is a rational function, which may have vertical and horizontal asymptotes.
A vertical asymptote occurs where the denominator of the function equals zero and the function is undefined. Setting the denominator of f(x) to zero, we get:
This gives us a vertical asymptote at x = 16.
The horizontal asymptote of a rational function can be found by comparing the degrees of the numerator and the denominator. Since the degrees are the same (both are first degree), the horizontal asymptote is the ratio of the leading coefficients. Hence, the horizontal asymptote is y = 4/1 = 4.
The end behavior of the function can be determined by investigating the signs of the function as x approaches infinity and negative infinity. As x approaches infinity or negative infinity, the function f(x) approaches the horizontal asymptote y = 4.