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  1. Let
    g(x) = 2- x
    3 + 4x
    and h(x) =7
    1. Find (g o h)(x) (don't forget to simplify).
    2. Find the inverse of g. Label your answer clearly either as "the inverse of g" or with "g-1(x) = ... .

    1. Find the domain of the function k defined by k (t) = (5 - t)1/2 + ln (t + 3) + 3t-2
    2. The graph of a function h is created by shifting the graph of y = 2x to the right 5 units, then reflecting it in the x-axis, and then shifting it up 10 units. Draw the graph of h and find h(x). Show and label with its intercept or equation any asymptote.

  2. Draw and name the graph of each of the following equations. On your graph label with their coordinates all vertices and intercepts. Label any asymptote with its equation. Indicate clearly the scale on all axes.
    1. x + 2y2 = 0
    2. 9x2 + y2 = 36

  3. Solve each of the following equations for x (exact values):
    1. 3 log5 (4x - 1) = 1/2
    2. 105 (85x+3) = 315

    1. Write as a single logarithm and simplify: 4 ln (s + 2) - 3 ln (s2 - 4) + ln (s + 1)
    2. Show the tail behavior and behavior at x = 2 for each of the following polynomials:
      1. f is a fifth-degree polynomial with five different x-intercepts including
        x = 2 and f(x) < 0 for x > 2
      2. g is an eighth-degree polynomial with x-intercept 2 and g(x) > 0 for x > 1

  4. For each of the following the answers must be exact.
    1. Write without logarithms and negative exponents: log4 (cube-root of 16) + e-3 ln x
    2. Use the properties of logarithms to simplify the following expression so that the result does not contain logarithms of products, quotients or powers. Assume x > 0.
      ln x2 (x + 1)1/2
      (x2 + 2x + 1)1/2

  5. Let
    g(x) = (x - 3)2
    (x + 4)(12 - 3x)
    1. Find (i) the x- and y-intercepts and (ii) the vertical and horizontal asymptotes for the graph of g. Label your answers clearly. If the graph of g does not have one of the intercepts or asymptotes write NONE for that item.
    2. Use a table of signs to solve the inequality
      (x - 3)2
      (x + 4)(12 - 3x)
      > 0

  6. Fill in each of the empty cells (those with no x's) in the following table. If a value does not exist write DNE in the cell. Notice that in the last column cos t = 5/7 with - p/2 < t < 0. >
    t = 3p
    angle with - p/2 < t < 0
    sin txxxxxxxxxx___________
    cos t__________________5/7
    tan txxxxx_________________
    sec txxxxxxxxxxxxxxx_____
    csc t_____xxxxxxxxxxxxxxx

    1. Convert the angle 780o to radians.
    2. In what quadrant does the terminal side of an angle of 5 radians lie? Justify your answer.
    3. Let sin u = 2/5 and p/2 < x < p. Find each of the following (exact values).
      1. cos (u + 7p/6)
      2. sin (u - 2p/3)

    1. Show that the expression
      (sin t - cos t)(sin t + cos t)
      2 sin t cos t
      can be simplified to a multiple of cot 2t.
    2. Find all solutions for x (exact values in radians): 2 cos 3x sin x + 31/2 cos 3x = 0

  7. Joel wants to determine the height of a hot air balloon that is sitting on the ground. Using a surveyor's transit that is 5 feet above the ground he stands at a point A and measures the angle of elevation of the top of the balloon as 16.3. He then walks 30 feet straight toward the balloon to a point B and again using the transit measures the angle of elevation of the top of the balloon as 18.5. Draw a picture illustrating this situation, marking clearly all known angles and distances and any variable(s) used in solving the problem. State clearly any equation(s) used (there must be at least one). Then find the height of the balloon (to the nearest foot). Give a complete answer (The height of the balloon is ...).