MATH 125 FALL 2018 ELAC TEST 3 TAKE HOME Name: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine whether the functions f and g are inverses of each other. 1) f(x) = (x + 2) 3-1; g(x) = 3 x + 1-2 1) For the given functions f and g, find the composition. 2) f(x) = x 2 + 3x - 1; g(x) = x Find (f g)(16). 2) Find the distance between the pair of points. 3) (2 7, -1) and (5 7, 0) 3) Solve. ) The hypotenuse of an isosceles right triangle is 7 feet longer than either of its legs. Find the exact length of each side. ) 5) 5x + = 3x + 1 + 3 5) 6) An isosceles right triangle has legs of equal length. If the hypotenuse is 10 inches long, find the length of each leg. 6) 7) A balloon is secured to rope that is staked to the ground. A breeze blows the balloon so that the rope is taut while the balloon is directly above a flag pole that is 50 feet from where the rope is staked down. Find the altitude of the balloon if the rope is 120 feet long. 7) 8) x - 1 = x - 9 8) 9) Scott set up a volleyball net in his backyard. One of the poles, which forms a right angle with the ground, is 6 feet high. To secure the pole, he attached a rope from the top of the pole to a stake 11 feet from the bottom of the pole. To the nearest tenth of a foot, find the length of the rope. 9) Solve the problem. 10) A vendor sells hot dogs, bags of potato chips, and soft drinks. A customer buys hot dogs, 2 bags of potato chips, and 5 soft drinks for $15.25. The price of a hot dog is $0.75 more than the price of a bag of potato chips. The cost of a soft drink is $2.25 less than the price of two hot dogs. Find the cost of each item. 10) 11) Anne and Nancy use a metal alloy that is 18.2% copper to make jewelry. How many ounces of an alloy that is 11% copper must be mixed with an alloy that is 25% copper to form 105 ounces of the desired alloy? 11) 1
12) A cruise boat traveled 96 miles downstream, with the current, in 2 hours. The return trip upstream, against the current, covered the same distance but took 6 hours. Find the average velocity of the current. 12) For the given one-to-one function f, find the following. 13) f(x) = x - 3-3 Find f(3) and f -1 (0). 13) Write an equation of the circle with the given center and radius. 1) (2, -); 13 1) Perform the indicated operation. Write the result in the form a + bi. 15) 5 + 2i 9 + 8i 15) Sketch the graph of the equation. If the graph is a parabola, find its vertex. If the graph is a circle, find its center and radius. 16) x 2 + (y - ) 2 = 16 16) 17) x = -(y - 5) 2 + 1 17) 18) y = -(x - 3)2 + 6 18) 19) (x - ) 2 + (y - 2) 2 = 16 19) 20) x = (y - 2) 2 + 20) Find the inverse of the one-to-one function. 5 21) f(x) = 3x - 1 21) Graph the exponential function. 22) f(x) = 2 x - 1 22) 23) f(x) = 1 3 x 23) Graph the piecewise defined function. 2) 5x - 2 if x 0 f(x) = 1 2 x - 5 if x > 0 2) 2
Solve the inequality. Graph the solution set and write the solution set in interval notation. 25) x 2-5x 9 25) 26) x - 7 x + 3 < 0 26) 27) x 2 27) x + 28) 3x x + 6 < x 28) Find an equation of the line. Write the equation using function notation. 29) Through (-5, -1); perpendicular to 5x + y = 60 29) Find the power of i. 30) (-3i) 7 30) Sketch the graph of the quadratic function. Give the vertex and axis of symmetry. 31) f(x) = - 1 5 (x + 2) 2-3 31) 32) f(x) = (x - 5)2 + 2 32) Sketch the graph of the quadratic function by finding the vertex, intercepts, and determining if the graph opens upward or downward. 33) f(x) = -2x 2 + 8x 33) 3) f(x) = x 2 + 8x +7 3) Graph the equation. 35) x2 25 + y2 9 = 1 35) 3
(x - 2)2 (y + 1)2 36) + 16 = 1 36) Find the quotient and simplify. 37) p2-5p + pq - 5q 12p 2-12q 2 p - 5 2p - 2q 37) Solve the system. 38) x + y + z = 2 x - y + 5z = -16 x + y + z = -1 38) Express as the logarithm of a single expression. Assume that variables represent positive numbers. 39) 7 log x + 1 8 log x - 2 log (x + 5) 39) Use the square root property to solve the equation. 0) (x - 12) 2 = -25 0) 1) (2x - 1)2 = 81 1) Simplify. 2) 6x-1 + (5y) -1 x -2 2) For the given functions f and g, find the requested function. 3) f(x) = 2x 3-3; g(x) = 6x 2 + 1 Find (f g)(x). 3) ) f(x) = 3x + 1; g(x) = 5x - 1 Find f (x). ) g Use the discriminant to determine the number and type of solutions of the equation. 5) x2 - x + 1 = 0 5) 6) -6-7x 2 = 5x - 7 6) Find the midpoint of the line segment whose endpoints are given. 7) (3 6, 6), (6 6, 9 6) 7) Solve the equation by completing the square. 8) 16x 2 + 1 = 3x 8)
Solve the equation. 9) log x = log 5 + log (x - 1) 9) 50) y + 3-6 y - 3 = 1 y 2-9 50) Graph the function. 51) f(x) = x + 7 + 3 51) 52) y = log 2 x 52) 53) y = log 1/3 x 53) Use the properties of exponents to simplify the expression. Write with positive exponents. 5) (-3p 3/5-7p /5 )(-3p 3/5-7p /5 ) 5) Graph the inverse of the function on the same set of axes. 55) 55) Add or subtract. Assume all variables represent positive real numbers. 56) 6 3 x 3 y 7-2xy 3 8y 56) Use Cramer's rule to solve the system. 57) -x + 5y - z = -22 x - 5y - 8z = -23-7x + y + z = -38 57) Find the equation of the line. Write the equation using standard notation. 58) Slope - 3 ; through (2, ) 58) 8 Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. 59) 5x + 2y + z = -11 59) 2x - 3y - z = 17 7x - y = 12 5
Multiply or divide. 60) -5 60) Use the quadratic formula to solve the equation. 61) 7x 2-11x - 3 = 0 61) 62) x(8x + 9) = 7 62) Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. 2x 63) 63) 5 1331x 17 y 13 6