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Exam 2, Math 1050 (100 pts total), Name: Show your work throughout the test.
(10 pts) 1. True or False. ( ) (a) If โ7 is a zero of the polynomial f (x), then the remainder is zero when one divides f (x) by x โ 7.
False ( ) (b) Suppose f is a function that has an inverse. Based on the chart (on the right), what is f โ^1 (3)?
x f (x) 1 4 2 3 3 2 4 0 Sorry, this isnโt even a True/False question. The answer is 2.
( ) (c) The functions f (x) = 3x + 1 and g(x) = xโ 3 1 are inverses. True. f (g(x)) = 3xโ 3 1 + 1 = x โ 1 + 1 = x, and g(f (x)) = x, so they are inverses.
(10 pts) 2. Fill in the blanks. (a) If a 2 ร 3 matrix is multiplied by a 3 ร 7 matrix, what is the size of the resulting matrix?
2 ร 7
(b) Evaluate the matrix product
(c) If logb 10 = 3, then logb (100) = logb (100) = logb(10^2 ) = 2 logb 10 = 2(3) = 6
(10 pts) 3. Draw a graph of f (x) = 2 โ log 3 x. Label at least three points and any asymptotes.
(10 pts) 4. Show how you can use the fact that log 10 4 = .6021 and log 10 7 = .8451 and log 10 9 = .9542 to compute log 10 634. log 10 634 = log 10 3222 ยท^7 = log 10 32 + log 10 7 โ log 10 22 = .9542 +. 8451 โ. 6021
(10 pts) 5. Solve log(8logx โx 7) = 2. Soln: Note log(8x โ 7) = 2 log x = log(x^2 ). Since log(ยท) is one-to-one, it means that 8x โ 7 = x^2. (Or just exponentiate both sides using base 10.) Therefore, x^2 โ 8 x + 7 = 0, and thus (x โ 7)(x โ 1) = 0. So x = 1, 7.
(10 pts) 6. Find the inverse of the function f (x) = โx โ 3. What are the domain and range of f (x) and f โ^1 (x)? Inverse is f โ^1 (x) = x^2 + 3. Domain of f = Range of f โ^1 = [3, โ). Range of f = Domain of f โ^1 = [0, โ).
(10 pts) 7. The amount (in grams) of a radioactive substance remaining t years from now is given by the formula Q = 42 (2โ.^017 t). After how many years will the amount remaining be 3 grams? Answer: Set 3 = 42 (2โ.^017 t). Then 141 = 2โ.^017 t, so ln
14
= โ. 017 t ln 2. This means t = ln(^
โ.017 ln 2.
