Exercises


  1. Write the following using set notation.
    1. The set of all natural numbers that are greater than or equal to \(4\).
    2. The set of all real numbers that are greater than or equal to \(-3\) and less than \(7\).
  2. Determine the domain and range of the following functions.
    1. \(\{(-2,3),~ (3,4),~ (8,5),~ (12, 6)\}\)
    2. 1 maps to negative 3, 4 maps to 5, 9 maps to negative 4, 16 maps to 5, and 7 maps to 11.
    3. A graph with the following points plotted: (negative 2, 0), (negative 1, 3), (0, 1), (1, 3), and (2, 0).
    4. An open dot at (negative 3, 0). A line segment from (negative 3, 0) to (0, 3) inclusive. An open dot at (4, 0) connected to a line that passes through (4, 0).
    5. Part of a parabola that opens upwards and has end points (negative 2, 6) and (4, 0) with vertex (2, negative 2).
    6. A line segment from (negative 3, negative 3) to (0, negative 3) where (0, negative 3) is an open dot. Another line segment exists between (0, 2) and (3, 2). There is an open dot at (3, 2).
  3. Determine the domain and range of the following linear relations.
    1. \(y=x+4\)
    2. \(y=5\)
    3. \(x=7\)
  4. Consider the linear function \(f(x)=-x+2\).
    1. If the domain of this function is \(\mathbb{R}\), determine the range.
    2. If the domain of this function is \(\mathbb{W}\), determine the range.
    3. If the domain of this function is restricted to \(\{x \in \mathbb{R} \mid x \lt -1\}\), determine the range.
    4. If the range of this function is \(\{y \in \mathbb{R} \mid y \le 4\}\), determine the domain.
  5. Determine the domain and range of the following quadratic functions.
    1. \(y=(x-4)^2+5\)
    2. \(y=-(x-1)^2-3\)
    3. \(y=(x-4)(x+2)\)
    4. \(y=-2(x+1)(x-1)\)
    1. Determine the vertex of \(f(x)=x^2+10x+16\) using the indicated method.
      1. Factoring.
      2. Completing the square.
      3. Using the vertex formula.
      4. Partial factoring.
    2. State the range.
  7. Consider the quadratic function \(f(x)=2x^2+3\).
    1. Determine the domain and range.
    2. If the domain is restricted to \(\{x \in \mathbb{R} \mid x \ge -2\}\), determine the range.
    3. If the domain is restricted to \(\{x \in \mathbb{R} \mid x \le -2\}\), determine the range.
  8. In a compressor, the pressure of air, measured in pounds per square inch (psi), is given by the function

    \(P(t)=\dfrac{25}8t^2-25t+100\)

    where \(t\) is the time the compressor is running, in minutes.

    While the compressor is being used, the pressure of air inside the compressor decreases. Once the compressor stops being used, the pressure of air inside the compressor increases until it reaches \(250 \) psi, at which time the compressor stops running. The function \(P(t)=\dfrac{25}8t^2-25t+100\) describes the pressure in the compressor from \(t=0\) until the compressor stops running.

    1. Determine \(P(0)\) and \(P(2)\).
    2. At what time is the pressure the lowest? What is the lowest pressure?
    3. At what time does the pressure reach \(250\) psi?
    4. Determine the domain of \(P(t)\).
    5. Determine the range of \(P(t)\).
    6. Sketch a graph of this function.
  9. Recall that the square root of a negative value is not a real number.
    1. Determine the domain of \(f(x)=\sqrt{x}\).
    2. Determine the domain of \(g(x)=\sqrt{x-5}\).
    3. Determine the domain of \(h(x)=\sqrt{(x-5)(x-1)}\). (Hint: When using set notation, the word 'or' can be used to indicate that either of two conditions can be satisfied for an element to be part of the set).