Final answer:
The function g(x) = 1/(x+1) - 4 has a vertical asymptote at x = -1 and a horizontal asymptote at y = -4. Its domain is all real numbers except x ≠ -1, and its range is all real numbers except y ≠ -4.
Step-by-step explanation:
The student asks about the asymptotes, domain, and range of the function g(x) = 1/(x+1) - 4. To identify the asymptotes, we look for values that make the function undefined. Since division by zero is undefined, the vertical asymptote occurs at the input value which makes the denominator zero, which is x = -1. This is a vertical asymptote.
For the horizontal asymptote, we recognize that as x goes to infinity, the term 1/(x+1) approaches zero. Thus, the function approaches y = -4, which is a horizontal asymptote.
The domain of the function is all real numbers except where the denominator is zero. Thus, the domain is x ≠ -1. The range of the function is all real numbers since the function can take any value except for the horizontal asymptote. Therefore, the range is y ≠ -4.