Test Your Knowledge: Piecewise Functions Algebra 2 Questions

📚 Quick Study Guide

• 🧮 A piecewise function is a function defined by multiple sub-functions, each applying to a certain interval of the main function's domain.
• 📈 To evaluate a piecewise function, first determine which interval of the domain the input value belongs to.
• ✏️ Then, use the corresponding sub-function to calculate the output value.
• 📐 Piecewise functions can be continuous or discontinuous. A continuous piecewise function has no breaks in its graph.
• 💡 When graphing, pay special attention to the endpoints of each interval. Use open or closed circles to indicate whether the endpoint is included in the interval.
• 📝 The general form for piecewise functions can be shown as: $f(x) = \begin{cases} f_1(x) & \text{if } x < a \\ f_2(x) & \text{if } a \leq x < b \\ f_3(x) & \text{if } x \geq b \end{cases}$
• 🧐 Pay close attention to the inequality symbols ($<, \leq, >, \geq$) to decide which piece of the function applies for any given $x$-value.

Practice Quiz

1. What is the value of the piecewise function $f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ 2x + 1 & \text{if } x \geq 0 \end{cases}$ when $x = -2$?
   1. -4
   2. -3
   3. 4
   4. -1
2. Which of the following piecewise functions is defined for all real numbers?
   1. $f(x) = \begin{cases} x + 1 & \text{if } x < 2 \\ x - 1 & \text{if } x > 2 \end{cases}$
   2. $f(x) = \begin{cases} x^2 & \text{if } x \leq 0 \\ \sqrt{x} & \text{if } x > 0 \end{cases}$
   3. $f(x) = \begin{cases} 2x & \text{if } x < 1 \\ 3x & \text{if } x > 1 \end{cases}$
   4. $f(x) = \begin{cases} x^3 & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}$
3. Given the piecewise function $g(x) = \begin{cases} 3 & \text{if } x \leq -1 \\ x + 4 & \text{if } -1 < x < 2 \\ 1 & \text{if } x \geq 2 \end{cases}$, what is $g(0)$?
   1. 3
   2. 4
   3. 1
   4. 0
4. What is the domain of the piecewise function $h(x) = \begin{cases} \frac{1}{x} & \text{if } x < -1 \\ x^2 & \text{if } x \geq 0 \end{cases}$?
   1. $(-\infty, -1) \cup (0, \infty)$
   2. $(-\infty, -1) \cup [0, \infty)$
   3. $(-\infty, \infty)$
   4. $(-1, 0)$
5. Evaluate the following piecewise function at $x = 5$: $f(x) = \begin{cases} x-2, & \text{if } x < 3 \\ 2x+1, & \text{if } 3 \leq x \leq 7 \\ 16, & \text{if } x > 7 \end{cases}$
   1. 3
   2. 11
   3. 16
   4. 7
6. Which of the following is an accurate description of the range of the piecewise function $f(x) = \begin{cases} -x, & \text{if } x < 0 \\ 0, & \text{if } 0 \leq x \leq 5 \\ x-5, & \text{if } x > 5 \end{cases}$?
   1. All real numbers
   2. All non-negative real numbers
   3. All non-positive real numbers
   4. The set {0}
7. For the function $f(x) = \begin{cases} x+2, & \text{if } x < -1 \\ x^2, & \text{if } -1 \leq x < 2 \\ 4, & \text{if } x \geq 2 \end{cases}$, is the function continuous at $x = -1$?
   1. Yes
   2. No
   3. Cannot be determined
   4. Only continuous from the left