Introduction to piecewise functions test questions and answers

📚 Quick Study Guide

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• A piecewise function is defined by multiple sub-functions, each applying to a specific interval of the main function's domain.
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• Each sub-function has its own formula and domain restriction (interval).
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• To evaluate a piecewise function, determine which interval the input value falls into and then use the corresponding sub-function.
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• Graphing piecewise functions involves plotting each sub-function over its specified interval. Pay close attention to endpoints; use open or closed circles to indicate whether the endpoint is included or excluded.
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• Discontinuities can occur at the boundaries between intervals if the sub-functions don't 'meet' smoothly.
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• The general form of a piecewise function is: $f(x) = \begin{cases} f_1(x), & \text{if } x \in D_1 \\ f_2(x), & \text{if } x \in D_2 \\ ... & ... \\ f_n(x), & \text{if } x \in D_n \end{cases}$ where $f_i(x)$ are the sub-functions and $D_i$ are the corresponding intervals.

🧪 Practice Quiz

1. What is a piecewise function?
1. A function with a constant slope.
2. A function defined by multiple sub-functions, each applying to a specific interval.
3. A function that is continuous everywhere.
4. A function that is always linear.
2. Which of the following is an example of a piecewise function?
1. $f(x) = x^2$
2. $g(x) = \begin{cases} x, & \text{if } x < 0 \\ x^2, & \text{if } x \geq 0 \end{cases}$
3. $h(x) = 2x + 1$
4. $k(x) = |x|$ for $x \geq 0$
3. Evaluate the piecewise function $f(x) = \begin{cases} x + 1, & \text{if } x < 1 \\ 3x, & \text{if } x \geq 1 \end{cases}$ at $x = 0$.
1. 0
2. 1
3. 3
4. Does not exist
4. Evaluate the piecewise function $f(x) = \begin{cases} x + 1, & \text{if } x < 1 \\ 3x, & \text{if } x \geq 1 \end{cases}$ at $x = 1$.
1. 0
2. 1
3. 3
4. 2
5. What is the domain of the sub-function in a piecewise function called?
1. Range
2. Interval
3. Slope
4. Y-intercept
6. At what points can discontinuities occur in a piecewise function?
1. Only at x = 0
2. Only when the sub-functions are linear
3. At the boundaries between intervals.
4. Piecewise functions cannot have discontinuities.
7. What is the range of the piecewise function $f(x) = \begin{cases} 2x, & \text{if } x < 0 \\ x^2, & \text{if } x \geq 0 \end{cases}$?
1. $(-\infty, \infty)$
2. $[0, \infty)$
3. $(-\infty, 0]$
4. $(0, \infty)$