College Algebra Questions
With Solutions and Explanations - sample 1

Solutions and full explanations to the set of 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(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 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 solutions. Solve the equation x2 + 1 = 1 for to get

    x = 0.

    The given equation has one 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)U(-2 , +∞) B. (-∞ , 1)U(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)U(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.

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