Exercises


  1. Complete the chart below.
     
    Interval Notation Inequality Notation English Sentence
    \( ( -\infty, -2) \)   The set of all real numbers less than \( -2 \).
      \( 1 \leq x \leq 10 \)  
        The set of all real numbers less than or equal to \( 6 \) or greater than \( 8 \).
    \( \left[ -2, 2 \right] \cup \left[ 6, \infty \right) \)    
  2. Use a number line to graph the intervals below.
    1. \( \left[ -3, 4 \right) \)
    2. \( \left( -\infty, 2 \right) \cup \left[ 3, \infty \right) \)
    3. \( x \geq 6 \) and \( x \leq 10 \)
  3. Piecewise function; y=-0.5x on [-infinity, 2]; y = x-3 on [2,infinity]
    Given the graph of \( y = f(x) \) shown, identify the following:
    1. the domain of the graph
    2. the range of the graph
    3. the increasing and decreasing intervals
    4. the intervals where \( f(x) \geq 0 \)
  4. Given the function \( y = -2x^2 + 24x + 26 \), determine the following:
    1. the local maxima and minima
    2. the \( x \) and \( y \) intercepts of the graph of the function \( y \)
    3. the sketch of the function and the interval(s) where \( y \leq 0 \)
    4. the interval(s) where the graph of \( y \) increases
  5. Given the graphs of \( y = f(x) \) and \( y = g(x) \) as shown in the graph below
    Quadratic with zeros x=-2,4, vertex (1,-2); cubic with zeros x=-2,4; intersection at (1,-2)
    identify the following.
    1. where \( f(x) = g(x) \)
    2. the interval(s) where \( g(x) \lt 0 \)
    3. the interval(s) where \( f(x) \geq g(x) \)
    4. the interval(s) where both functions are decreasing.
    1. Graph \( y = 2(x + 3)(x - 5) \)
    2. Solve \( 2(x + 3)(x - 5) \lt 0 \)
    1. Graph \( y = 10 + x - 2x^2 \)
    2. Solve \( 10 + x - 2x^2 \lt 0 \)