Honors Algebra 2: Piecewise Functions Worksheet 1

Given

\( f(x) = \begin{cases} 3x - 7, & \text{if } x \leq 2 \\ 6 - 2x, & \text{if }x > 2 \end{cases}\) \(g(x) = \begin{cases} 3x + 5, & \text{if } x < 5 \\ -x + 3, & \text{if } x \geq 5 \end{cases} \) \( h(x) = \begin{cases} \displaystyle \frac{2}{3}x + 1, & \text{if } x < -3 \\ 2x - 3, & \text{if } x \geq -3 \end{cases} \)
  1. Find \( f(0) \).
    2. Solution -7
  2. Find \( f(2) \).
    3. Solution -1
  3. Find \( f(4) \).
    4. Solution -2
  4. Find \( f(-3) \).
    5. Solution -16
  5. Find \( g(5) \).
    6. Solution -2
  6. Find \( g(-4) \).
    7. Solution -7
  7. Find \( g(3) \).
    8. Solution 14
  8. Find \( g(10) \).
    9. Solution -7
  9. Find \( h(-9) \).
    10. Solution -5
  10. Find \( h(-3) \).
    11. Solution -9
  11. Find \( h(6) \).
    12. Solution 9
  12. Find \( h(1) \).
    13. Solution -1
  13. Graph \( f(x) = \begin{cases} 3, & \text{if } x \leq 4 \\ -1, & \text{if } x > 4 \end{cases} \)
    14. Solution graph
  14. Graph \( f(x) = \begin{cases} x - 4, & \text{if } x < 2 \\ 3 - x, & \text{if } x \geq 2 \end{cases} \)
    15. Solution graph
  15. Graph \( f(x) = \begin{cases} x + 1, & \text{if } x < 0 \\ -x + 1, & \text{if } 0 \leq x \leq 2 \\ x - 2, & \text{if } x > 2 \end{cases} \)
    16. Solution graph
  16. Graph \( f(x) = \begin{cases} 2x, & \text{if } x \geq -1 \\ 3x, & \text{if } -2 < x < -1 \\ -x, & \text{if } x \leq -2 \end{cases} \)
    17. Solution graph
  17. Graph \( f(x) = \begin{cases} 2, & \text{if } x \leq -3 \\ -1, & \text{if } -3 < x < 3 \\ 3, & \text{if }x \geq 3 \end{cases} \)
    18. Solution graph
  18. Write the piecewise function whose graph would look like the graph below.
    graph
    19. Solution \( f(x) = \begin{cases} (x - 1) - 2, & \text{if } x > 1 \\ -2x + 2, & \text{if }x \le 1 \end{cases} \)
  19. Write the piecewise function whose graph would look like the graph below.
    graph
    20. Solution \( f(x) = \begin{cases} (x - 1) - 2, & \text{if } x \ge 1 \\ -2x + 2, & \text{if } x < 1 \end{cases} \)
  20. Write the piecewise function whose graph would look like the graph below.
    graph
    21. Solution \( f(x) = \begin{cases} (x - 1) - 2, & \text{if } x > 1 \\ -1, & \text{if } x = 1 \\ -2x + 2, & \text{if }x < 1 \end{cases} \)
  21. A company provides bus tours of historical cities. The given function describes the rate for small groups and the discounted rate for larger groups, where \(x\) is the number of people in your group:
    \( C(x) = \begin{cases} 8.95x, & \text{if } x \in \mathbb{Z} ; 0 < x \leq 10 \\ 7.50x, & \text{if } x \in \mathbb{Z} ; x > 10 \end{cases}\)
    1. Graph the function.
      2. Solution
    2. Identify the domain of the function
      3. Solution \( \{ x \mid x \in \mathbb{Z}; x > 0 \} \)
  22. You are employed by a company in which commission rates are based on how much you sell. If you sell up to $100,000 of merchandise in a month, you earn 5% of sales as a commission. If you sell over $100,000, you earn 8% commission on your sales. Write a piecewise function that gives the amount you earn in commission in a given month for \(x\) dollars in sales.
    23. Solution \( C(x) = \begin{cases} 0.05x, & \text{if } 0 < x \le 100,000 \\ 0.08x, & \text{if }x > 100,000 \end{cases} \)