Solutions and Explanations to Questions College Algebra - sample 1

Solutions and full explanations to the college algebra multiple choice questions are presented.

  1. 9log9(4) =
    Solution
    Exponential and log functions are inverse of each other. Hence
    aloga(x) = x , for all x real and positive.
    and therefore
    9log9(4) = 4
  2. 3log3(-5) =
    Solution
    Since -5 is not in the domain of function log3(x),
    3log3(-5) is undefined
  3. If f(x) = -2x2 + 8x - 4, which of the follwoing is true?
    A. The maximum value of f(x) is - 4.
    B. The graph of f opens upward.
    C. The graph of f has no x-intercept
    D. f is not a one to one function.
    Solution
    f(x) is a quadratic function and its graph is a parabola that may be intercepted by horizontal lines at two points and therefore is not a one to one function. The answer is D
  4. If f(x) = 5 - 2x, then f -1(-3) =
    Solution
    Find f -1(x) and then Find f -1(- 3)
    y = 5 - 2x , given
    x = 5 - 2y , interchange x and y
    2y = 5 - x , y = log2(5 - x) , solve for y
    f -1(x) = log2(5 - x) , inverse function
    f -1(- 3) = log2(5 -(- 3)) = log2 8
    = log2(23) = 3
  5. If logx(3) = 1/4, then x =
    Solution
    Rewrite the given equation in exponential form
    logx(3) = 1/4 if and only if x(1/4) = 3
    We now solve, for x, the exponential equation obtained above by raising both sides to the power 4.
    (x(1/4)) 4 = 3 4
    x = 3 4 = 81
  6. If f(x) = -x2 + 1, then f(x + 1) =
    Solution
    Substitute x by x + 1 in the formula of f(x) to obtain f(x + 1).
    f(x + 1) = - (x + 1) 2 + 1
    Expand and simplify.
    f(x + 1) = - x 2 - 2x - 1 + 1 = - x 2 - 2x
  7. If f(x) = x - 4, then (f o f)(3) =
    Solution
    (f o f)(3) = f(f(3)) = f(3 - 4) = f(-1) = - 5
  8. If ln(3x - 2) = 1, then x =
    Solution
    Rewrite given equation in exponential form.
    ln(3x - 2) = 1 if and only if e 1 = 3x - 2
    Solve e 1 = 3x - 2 for x.
    x = (e + 2) / 3
  9. The number of real solutions of (x2 + 1)2 + 2(x2 + 1) - 3 = 0 is equal to
    Solution
    Let u = x2 + 1 and rewrite the given equation in terms of u as follows
    u 2 + 2u - 3 = 0
    Factor and solve the above equation
    (u + 3)(u - 1) = 0
    two solutions: u = x2 + 1 = - 3 and u = x2 + 1 = 1
    Equation x2 + 1 = - 3 has no real solutions. Solve the equation x2 + 1 = 1 for to get
    x = 0.
    The given equation has one real solution.
  10. If the graph of y = (x - 2)2 - 3 is translated 5 units up and 2 units to the right, then the equation of the graph obtained is given by
    Solution
    If the graph of y = f(x) is translated 5 units up, the equation of the new graph is given by
    y = f(x) + 5
    If the graph of y = f(x) + 5 is translated 2 units to the right, the equation of the new graph is given by
    y = f(x - 2) + 5 = ((x - 2) - 2)2 - 3 + 5
    = (x - 4)2 + 2
  11. If f(x) = -ex - 2, then the range of f is given by the interval
    A. (-∞ , -2)
    B. (-∞ , +∞)
    C. (-2; , +∞)
    D. (-∞ , +2)
  12. If f(x) = √(x - 1) / (x2 - 9), then the domain of f is given by the interval
    A. (1 , +∞)
    B. (-3 , +3)
    C. [1 , 3)U(3 , +∞)
    D. (-3 , 3)U(3 , +∞)
  13. The number of points of intersections of the graphs of y = 2x and y = -x2 + 2 is equal to
    A. 0
    B. 1
    C. 2
    D. 3
  14. If f(x) = ln(x + 1) - 2, then f-1(x) =
    A. ex + 1 - 2
    B. ex - 2
    C. ex + 2 - 2
    D. ex + 2 - 1
  15. For all x real, √(x2 -4x + 4) =
    A. x - 2
    B. x + 2x + 2
    C. |x - 2|
    D. x + 2
  16. The value of x that makes x2 + 6x + 13 maximum is equal to
    A. 6
    B. -3
    C. 13
    D. 3
  17. eln(3) - ln(2) + ln(1/x) =
    A. 3 / (2x)
    B. 3x/2
    C. 1 + 1/x
    D. 3/2 - 1/x
  18. If f(x) = (x - 1) / (x + 2), then the range of f is given by the interval
    A. (-∞ , -2) ∪ (-2 , +∞)
    B. (-∞ , 1) ∪ (1 , +∞)
    C. (-2; , +∞)
    D. (-∞ , 1)
  19. ln((x - 1)2) = 2 ln(x - 1) for all x in the interval
    A. (-∞ , +∞)
    B. [0 , +∞)
    C. (-∞ , 1) ∪ (1 , +∞)
    D. (1 , +∞)
  20. Let f(x) = x2 + 2x + 4. Which of the following statements is NOT true?

    A. f(x) has a maximum value
    B. The graph of f is not a line<
    C. The graph of f has no x-intercepts.
    D. The graph of f has a y-intercept.

    Answers to the Above Questions

    1. B
    2. D
    3. D
    4. C
    5. A
    6. B
    7. C
    8. B
    9. A
    10. D
    11. A
    12. C
    13. C
    14. D
    15. C
    16. B
    17. A
    18. B
    19. D
    20. A

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