Math 165, Chapter 4 (Solutions)

Math 180 - Reviewing Chapter 4 Name Solution If the following defines a one-to-one function, find the inverse. {(-13, 18), (18, 15), (-12, -4)} {(-13, 18), (18, 15), (-12, -4)} Decide whether or not the functions are inverses of each other. f(x) = 4x + 7 g(x) = x/4 - 7 f(g(x)) = 4(x/4 - 7) + 7 = x - 28 + 7 = x - 21 Not an Inverse Determine the inverse of the given function. {(-3, 4), (-1, 5), (0, 2), (4, 2), (7, 5)} {(-3, 4), (-1, 5), (0, 2), (4, 2), (7, 5)} The graph of a one-to-one function f is given. Draw the graph of the inverse function f⁻¹ as a dashed line or curve. f(x) = x³ + 4 x f(x) f⁻¹(x) -2 -5 -5 -2 -1 2 2 -1 0 4 4 0 2 10 10 2 [Graph with f(x) and f⁻¹(x) plotted] Use the graph of the given one-to-one function to sketch the graph of the inverse function. For convenience, the graph of y=x is also given. 1   [Graph with f(x) and y=x plotted] x f⁻¹(x) -2 -4 -1 -2 1 0 2 2 4 1 The function f is one-to-one. Find its inverse. f(x) = 3x² + 2; x ≤ 0 y = 3x² + 2 x = ±√(y-2)/3 x² = (y-2)/3 f⁻¹(x) = -√(x-2)/3 ; y ≥ 0 Determine the equation for the inverse function of y = (x+2)³ - 8. y = (x+2)³ - 8 y + 8 = (x+2)³ ∛(y+8) = x+2 ∛(y+8) - 2 = x f⁻¹(x) = ∛(x+8) - 2 Solve the problem. The profit P for selling items is given by the equation (P(x) = 2x - 500). Express the sales amount x as a function of the profit P. (2x = P + 500) (x = \frac{P + 500}{2} \Rightarrow X(P) = \frac{P + 500}{2}) Approximate each value using a calculator. Express answer rounded to three decimal places. (44\pi = 105.067) Solve the problem. A rumor that there will be an open book test on chapter 4 has spread in our class of 28 students. The mathematical model for the situation is (N = 28(11 - e^{-0.16d})) where N is the number of students who have heard the rumor and d is the number of hours that have elapsed since the rumor began. a) How many students will have heard the rumor after 5 hours?2 (N = 28(1 - e^{-0.16 \cdot 5})) (N = 28(1 - e^{-0.8})) (N \approx 15) students   b) 21 students have heard the rumor. How many hours ago did the rumor begin? (21 = 28(1 - e^{-0.16d})) (\frac{21}{28} = 1 - e^{-0.16d}) (\frac{3}{4} = 1 - e^{-0.16d}) (e^{-0.16d} = 1 - \frac{3}{4}) (e^{-0.16d} = \frac{1}{4}) (\ln(e^{-0.16d}) = \ln(\frac{1}{4})) (-0.16d = \ln(0.25)) (d = \frac{\ln(0.25)}{-0.16}) (d \approx 9) hours Graph the function. Graph the function (f(x) = 2x - 1) (Please See page 8) Answer the question. Define the number e. A) The number approximately equal to 2.72. B) The number defined by (e = \lim_{n \to \infty}(1 + \frac{1}{n})^n) in Calculus. C) The number that the expression, ((1 + \frac{1}{n})^n) approaches as n increases without bound. D) All of the above. [Box around D] Solve the equation. (9^{2x} \cdot 27^{(3-x)} = \frac{1}{9}) ((3^2)^{2x} \cdot (3^3)^{(3-x)} = 3^{-2}) (3^{4x} \cdot 3^{9-3x} = 3^{-2}) (3^{4x + 9 - 3x} = 3^{-2}) (4x + 9 - 3x = -2) (x = -11) (2(x^2 - 3) = 64) (2^{x^2 - 3} = 2^6) (x^2 - 3 = 6) (x^2 = 9) (x = \pm 3) Use a graphing calculator to solve the equation. Round your answer to two decimal places. (e^x - \ln x = 4) (Y_1 = e^x - \ln x) (Y_2 = 4) 2nd Trace Intersect (x = 1.48) Solve the given exponential equation. Round answer to three decimal places. (5^{2x} + 5(x+1) - 24 = 0) (Y_1 = 5^{2x} + 5^{x+1} - 24) (Y_2 = 0) 2nd Trace Intersect (x = 0.683) Change exponential expressions (2 \cdot 3^{x+5} = 15) (3^{x+5} = \frac{15}{2}) (\log_3(\frac{15}{2}) = x + 5) (x = \log_3(\frac{15}{2}) - 5) (x \approx -1.75) Convert to logarithmic form. (e^x = 15) (\ln 15 = x) Convert to exponential form. (\log_5(\frac{1}{25}) = -2) (5^{-2} = \frac{1}{25})