2.1 Piecewise Functions
Contents
Objectives:[edit]
- Evaluate piecewise functions
- Graph piecewise functions
- Create piecewise functions from a word problem or a graph
Definitions:[edit]
Note:[edit]
We begin with the Interlude: Introduction to Piecewise Functions. Piecewise functions often prove troubling to students throughout the semester. Being especially thorough with the introductory material should be helpful.
Lesson Guide[edit]
Warm-Up[edit]
Have students do the interlude and Problems 1 and 2. If you like, you can go through it together as a class or show them a similar example first. Alternatively, you can just have them work their way through the problems on their own and then discuss as a class what piecewise functions are. Many of your students will report that they don't know how to graph the function. If this is the case then walk them through plotting points and connecting those points with reasonable curves.
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Instructor Example |
Suggested Example |
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This is a sentence that is in place to generate space in the table |
Students will be uncomfortable with piecewise function notation. Take a few minutes to interpret what this seemingly strange notation means. One way to do this is the following:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = x^2-2 \text{ if } x<1 \text{, and}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = x+2 \text{ if } x\geq 1. } To save space (and be lazier...) we use a curly bracket: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)= \begin{cases} x^2 & \text{ for } x< 1 \\ x+2 & \text{ for } 1\leq x \end{cases} } Note: We may use the words ``for," ``if," ``when," or nothing at all before writing the domain of each piece. }} Note: Students may also struggle with the last part of the interlude, which has them evaluate a piecewise function at specific Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -values. Make sure to emphasize to students that the new piecewise function, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)} , is what we built out of the two pieces. When Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x<1} , we use the rule given by the first line of the piecewise function, and when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \geq 1} , we use the second line. Optional: Do this process as a physical demonstration. Come in with graphs for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=x^2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=x+2} on separate sheets of paper (each piece could be a different color). Make sure the scales of each of the graphs match. Then physically cut each graph and paste them together correctly. Be sure to actually draw the final answer with an open and closed circle separately on the board with open and closed circles (students can then copy the finished version into their notes). The doc cam would be a useful tool in this case.
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Evaluate piecewise functions[edit]
Demonstrate how to evaluate a piecewise function at a given input.
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Instructor Example |
Suggested Example |
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This is a sentence that is in place to generate space in the table |
Again, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)} be the function that is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-2} when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x<1} and is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+2} when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\geq 1} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)= \begin{cases} x^2-2 & \text{ for } x< 1 \\ x+2 & \text{ for } 1\leq x \end{cases}. } Someone might as you to evaluate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(0)} . In order to do this evaluation we note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 < 1} hence we need to plug zero into the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-2} . Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(0)=0^2-2=-2} . Similarly, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(2)=2+2} . |
Have students do Problem 3.
Highlight the domain of each of the functions and also carefully explain what the open circle and closed circle notation means on the graph of these functions (i.e., how this corresponds to the domains of the function pieces).
Graph piecewise functions[edit]
Have students do Problems 4 and 5. Be sure to tell students that there are places on Problem 5 where the function is not even defined.
Create piecewise functions from a word problem or a graph[edit]
Ask students to try and solve problems 6-8. Students may resist at first but there should not be any issue with letting them try their hand at the Problems without an explicit example first.
Have students do Problems 6-8.
Note: Problem 7 has some major issues. However, we have chosen not to fix problem 7 because we think these issues can be instructive if you know about them ahead of time. In particular, the ticket function is not defined for values between 74 mph and 75 mph. Make sure that you point this out to your students and use this to explain that sometimes in the real world people say things one way, but really mean something different. Ask students how they might fix the problem and use this as an opportunity to talk about where this problems breaks down.
Groups that are moving along well can continue by working on Problems 9 and 10 but not everybody will get this far. Tell students that the important thing is that they get through Problem 8.
