Final answer:
To find the domain, vertical asymptote, and x-intercept of the logarithmic function h(x) = -log₃ (x+4) - 2, we need to consider the restrictions on the input variable x. The domain is x > -4, the vertical asymptote is x = -4, and the x-intercept is x = 4/3.
Step-by-step explanation:
To find the domain of the logarithmic function h(x) = -log₃(x+4) - 2, we need to consider the restrictions on the input variable x. Since the logarithm function is only defined for positive values, we need to find the values of x that make x+4 positive. Solving the inequality x+4 > 0, we find that the domain of the function is x > -4.
The vertical asymptote of a logarithmic function is a vertical line that the graph approaches but never passes. In this case, the vertical asymptote occurs when the argument of the logarithm approaches zero. Therefore, by setting x+4 = 0, we find that the vertical asymptote is x = -4.
The x-intercept of a function is the point where it intersects the x-axis. To find the x-intercept of h(x), we set h(x) = 0 and solve for x. So, -log₃(x+4) - 2 = 0. Solving this equation, we find that the x-intercept is x = 4/3.
Based on the domain, vertical asymptote, and x-intercept, we can sketch the graph of the function h(x) = -log₃(x+4) - 2.