Honors Algebra 2: Piecewise Functions Worksheet 2

  1. Given \( g(x) = \begin{cases} -x + 12, & \text{if } x < 8 \\ \displaystyle \frac{1}{2}x - 2, & \text{if }x \ge 8 \end{cases} \)
    1. Find \(g(-2)\)
      2. Solution 14
    2. Find \(g(10)\)
      3. Solution 3
    3. Find \(g(8)\)
      4. Solution 2
    4. Which is greater, \(g(7.9)\) or \(g(8.1)\)?
      5. Solution \(g(7.9)\)
  2. Draw the graph of \( f(x) = \begin{cases} 1, & \text{if } x < 0 \\ 0, & \text{if } x = 0 \\ -1, & \text{if }x > 0 \end{cases} \)
    3. Solution
  3. Given \( h(x) = \begin{cases} x + 2, & \text{if } x \le 4 \\ 6, & \text{if }x > 5 \end{cases} \)
    1. Draw the graph
      2. Solution
    2. Find \(h(0)\)
      3. Solution 2
    3. Find \(h(4)\)
      4. Solution 6
    4. Find \(h(4.5)\)
      5. Solution Does Not Exist
    5. Find \(h(10)\)
      6. Solution 6
  4. Given \( f(x) = \begin{cases} 3x + 5, & \text{if } \displaystyle x < \frac{1}{2} \\ 2x + 1, & \text{if } \displaystyle x > \frac{1}{2} \end{cases}\)
    1. Find \( f(3)\)
      2. Solution 7
    2. Find \( \displaystyle f \left( \frac{1}{2} \right) \)
      3. Solution Does Not Exist
    3. Find \( \displaystyle f \left( \frac{1}{3} \right) \)
      4. Solution 6
    4. Find \( \displaystyle f \left( \frac{5}{2} \right) \)
      5. Solution 6
  5. Given \( g(x) = [x] \) (that is, the greatest integer function)
    1. Find \(g(3.2)\)
      2. Solution 3
    2. Find \(g(1.8)\)
      3. Solution 1
    3. Find \(g(-2.4)\)
      4. Solution -3
    4. Find \(g(-6.9)\)
      5. Solution -7
  6. Given \( h(x) = 3[x + 2] - 1 \) (where \([x]\) is the greatest integer function)
    1. Find \(h(1.8)\)
      2. Solution 8
    2. Find \(h(3.1)\)
      3. Solution 14
    3. Find \(h(-0.4)\)
      4. Solution 2
    4. Find \(h(-3.1)\)
      5. Solution -7
  7. Calculators cost $17.95 each if 1 - 10 are purchased, $15.95 each if 11 - 200 are purchased, and $14.50 each if more than 200 are purchased.
    1. Write a piecewise function \(C\) for the cost of \(x\) calculators.
      2. Solution \( C(x) = \begin{cases} 17.95x, & \text{if } x \in \mathbb{Z}; 0 < x \le 10 \\ 15.95x, & \text{if } x \in \mathbb{Z}; 10 < x \le 200 \\ 14.50x, & \text{if } x \in \mathbb{Z}; x > 200 \end{cases} \)
    2. Find \(C(17)\)
      3. Solution $271.15
    3. Find \(C(345)\)
      4. Solution $5,002.50
  8. A gift shop sells pewter mugs for $35. They are currently running an engraving promotion. The first six letters are engraved free. Each additional letter costs 20 cents.
    1. Write a piecewise model that gives the price of the mug with \(x\) engraved letters
      2. Solution \( C(x) = \begin{cases} 35, & \text{if } x \in \mathbb{Z}; 0 < x \le 6 \\ 35 + 0.2(x - 6), & \text{if }x \in \mathbb{Z}; 6 < x \end{cases} \)
    2. Graph the function
      3. Solution
    3. What is the price of a mug with the name Jamie Lynn Krane engraved?
      4. Solution $36.60