523 81. Financial Planning The traditional IRA (individual retirement account) is a common tax-deferred savings plan in the United States. Earned income deposited into an IRA is not taxed in the current year, and no taxes are incurred on the interest paid in subsequent years. However, when the money is withdrawn from the account after age 59 1 2 , taxes must be paid on the entire amount withdrawn. Suppose we deposited $5000 of earned income into an IRA, we can earn an annual interest rate of 4%, and we are in a 25% tax bracket. (Note: Interest rates and tax brackets are subject to change over time, but some assumptions must be made to evaluate the investment.) Also, suppose that we deposit the $5000 at age 25 and withdraw it at age 60, and that interest is compounded continuously. (a) How much money will remain after we pay the taxes at age 60? (b) Suppose that instead of depositing the money into an IRA, we pay taxes on the money and the annual interest. How much money will we have at age 60? (Note: We effectively start with $3750 (75% of $5000), and the money earns 3% (75% of 4%) interest after taxes.) (c) To the nearest dollar, how much additional money will we earn with the IRA? (d) Suppose we pay taxes on the original $5000 but are then able to earn 4% in a tax-free investment. Compare the balance at age 60 with the IRA balance. 82. Consider ƒ1x2 = log4 12x 2 - x2. (a) Use the change-of-base theorem with base e to write log4 12x 2 - x2 in a suitable form to graph with a calculator. (b) Graph the function using a graphing calculator. Use the window 3-2.5, 2.54 by 3-5, 2.54. (c) What are the x-intercepts? (d) Give the equations of the vertical asymptotes. (e) Why is there no y-intercept? 1. Consider the function ƒ1x2 = 23 2x - 7. (a) What are the domain and range of ƒ? (b) Explain why ƒ-1 exists. (c) Write an equation for ƒ-11x2. (d) What are the domain and range of ƒ-1? (e) Graph both ƒ and ƒ-1. How are the two graphs related with respect to the line y = x? 2. Match each equation with its graph. (a) y = log1/3 x (b) y = e x (c) y = ln x (d) y = a 1 3b x Chapter 4 Test A. x y 0 B. x y 0 C. x y 0 D. x y 0 3. Solve a 1 8b 2x-3 = 16x+1. CHAPTER 4 Test
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