Solutions and Explanations to Questions College Algebra - sample 1

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

1. 
   9^log₉(4) =
   Solution
   Exponential and log functions are inverse of each other. Hence
   a^log_a(x) = x , for all x real and positive.
   and therefore
   9^log₉(4) = 4
   
2. 
   3^log₃(-5) =
   Solution
   Since -5 is not in the domain of function log₃(x),
   3^log₃(-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 - 2^x, then f ^-1(-3) =
   Solution
   Find f ^-1(x) and then Find f ^-1(- 3)
   y = 5 - 2^x , given
   x = 5 - 2^y , interchange x and y
   2^y = 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 log_x(3) = 1/4, then x =
   Solution
   Rewrite the given equation in exponential form
   log_x(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) = -e^x - 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 = 2^x 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. e^{x + 1} - 2
    B. e^x - 2
    C. e^{x + 2} - 2
    D. e^{x + 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. 
    e^{ln(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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