Final answer:
The graph of y=log₃(x+2) has a domain of (-2, ∞), range of (-∞, ∞), and an asymptote at x=-2. For the compound interest problem, the future balance after 6 years with continuous compounding at a rate of 6.2% is about $69,211.77, and it takes approximately 11.16 years for the initial investment to double.
Step-by-step explanation:
Graphing the Logarithmic Function and Compound Interest Calculation
For graphing y=log₃(x+2), the domain can be determined by considering the argument x+2>0. This gives us x>-2, so our domain is (-2, ∞). The range is all real numbers, (-∞, ∞). The vertical asymptote is at x=-2 because the logarithm is undefined for zero and negative inputs. Three distinct points could be (1,0), (4,1), and (-1,-log₃(2)) or approximately (-1,-0.631).
The balance of an investment with continuous compounding can be found using the formula A=Pe^(rt), where P is the principal, r is the rate, and t is the time in years. Substituting the given values: A=43,000e^(0.062×6) results in a balance of approximately $69,211.77.
To find the time necessary to double the investment, using the continuous compounding formula, set A to double the principal, 2P: 2×43,000=43,000e^(0.062t). Solving for t gives a value of approximately 11.16 years.