#$&* course Mth 158 3/5/13 Around 12:00PM 035. * 35
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Given Solution: * * What did the horizontal line test tell you for this function? For every horizontal below the 'peak' of this graph the graph will intersect the horizontal line in two points. This function is not one-to-one on the domain depicted here. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: * 6.2.45 / 6.2.43 / 7th edition 5.2.28 looks like cubic thru origin, (1,1), (-1,-1), sketch inverse. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The inverse of one-to-one graph (x , y) will contain all points (y , x); therefore a graph containing points (2 , 1), (1 , 1) and (-1 , -1) will contain the exact same points last two points and (1 , 2). The graph is similar to the graph of x^3 when crossing the origin and the inverse is flipped along the line y = x. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: * * Describe your sketch of the inverse function. The graph of the function passes through (0, 0), (1,1), and (-1,-1). The inverse function will reverse these coordinates, which will give the same three points. Between x = -1 and x = 1 the graph of the original function is closer to the x axis than to the y axis, and is horizontal at the origin. The graph of the inverse function will therefore be closer to the y axis than to the x axis for y values between -1 and 1, and will be vertical at the origin. For x < 1 and for x > 1 the graph lies closer to the y axis than to the x axis. The graph of the inverse function will therefore lie closer to the x axis than to the y axis for y < 1 and for y > 1. In the first quadrant the function is increasing at an increasing rate. The inverse function will therefore be increasing at a decreasing rate in the first quadrant. In the third quadrant the function is increasing at a decreasing rate. The inverse function will therefore be increasing at an increasing rate in the third quadrant. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: * 6.2.36 / 6.2.32 / 7th edition 5.2.32 f = 2x + 6 inv to g = 1 / 2 * x - 3. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: f(g(x)) = 2((1/2)x - 3) + 6 = x - 6 + 6 = x g(f(x)) = (1/2)(2x + 6) - 3 = x + 3 - 3 = x confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: * * Show that the functions f(x) and g(x) are indeed inverses. f(g(x)) = 2 g(x) + 6 = 2 ( 1 / 2 * x - 3) + 6 = x - 6 + 6 = x. g(f(x)) = 1 / 2 * f(x) - 3 = 1/2 ( 2 x + 6) - 3 = x + 3 - 3 = x. Since f(g(x)) = g(f(x)) = x, the two functions are inverse. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: * 6.2.54 / 6.2.52 / 7th edition 5.2.44. inv of x^3 + 1; domain range etc.. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x = y^3 + 1 -y^3 = 1 - x y^3 = x - 1 cuberoot(y^3) = cuberoot(x - 1) y = cuberoot(x - 1) Domain: All real numbers Range: All real numbers confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: * * Give the inverse of the given function and the other requested information. The function is y = x^3 + 1. This function is defined for all real-number values of x and its range consists of all real numbers. If we switch the roles of x and y we get x = y^3 + 1. Solving for y we get y = (x - 1)^(1/3). This is the inverse function. We can take the 1/3 power of any real number, positive or negative, so the domain of the inverse function is all real numbers. Any real-number value of y can be obtained by using an appropriate value of x. So both the domain and range of the inverse function consist of all real numbers. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: * 6.2.66 / 6.2.64 / 7th edition 5.2.56. inv of f(x) = (3x+1)/(-x). Domain and using inv fn range of f. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x = (3y + 1) / (-y) -yx = 3y + 1 -3y - yx = 1 y(-3 - x) = 1 y = - (1) / (3 - x) Doman {x | x≠3} Range {All real numbers} confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: * * What is the domain of f? What is the inverse function? What does the inverse function tell you about the range of f? f(x) is defined for all x except x = 0, since division by 0 is not defined. If we switch x and y in the expression y = (3x + 1) / (-x) we get x = (3y + 1) / (-y). To solve for y we first multiply by -y, noting that this excludes y = 0 since multiplication of both sides by 0 would change the solution set. We get -y * x = -y * (3 y + 1) / (-y). The left-hand side simplifies to - x y and the right-hand side to 3 y + 1, giving us -x y = 3y + 1. Subtracting 3 y from both sides we get -x y - 3 y = 1. Factoring y out of the left-hand side (which becomes (-x - 3) * y; if you multiply this out you see that it is the same as -x y - 3 y) we get (-x - 3) y = 1, and dividing both sides by (-x - 3), which excludes x = -3, we get y = -1 / (x + 3). The domain of this function is the set of all real numbers except 3. Since the domain of the inverse function is the range of the original function, the range of the original function consists of all real numbers except 3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Ah, didn't catch the range things. Get it now. ------------------------------------------------ Self-critique Rating: 3 ********************************************* Question: * 6.2.96 / 6.2.94 / 7th edition 5.2.74. T(L) = 2 pi sqrt ( L / 32.2). Find L(T). YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: T(L) = 2*pi*sqrt( L / 32.2) T = 2*pi*sqrt( L / 32.2) T / 2pi = sqrt( L / 32.2) (T / 2pi)^2 = sqrt( L / 32.2)^2 (T^2) / (4*pi^2) = L / 32.2 L(T) = (T^2*32.2) / (4*pi^2) confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: * * We solve T = 2 pi sqrt( L / 32.2) for L. First squaring both sides we obtain T^2 = 4 pi^2 * L / 32.2. Multiplying both sides by 32.2 / ( 4 pi^2) we get L = T^2 * 32.2 / (4 pi^2). So our function L(T) is L(T) = 32.2 T^2 / (4 pi^2). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): That one was kind of fun. ------------------------------------------------ Self-critique Rating: 3 " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!