Difference between revisions of "1.6 Function Notation Input & Output"
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| − | Note:Contrast language here with previous example. That is emphasize that we're using words like ``find" and ``solve". Make sure your input arrows point towards the number in parentheses and not to the entire expression <math>c(\frac{1}{2})</math>. You may also note that <math>(\frac{1}{2})^2-3(\frac{1}{2})</math> is another way of writing the output. | + | Note:Contrast language here with previous example. That is emphasize that we're using words like ``find" and ``solve". Make sure your input arrows point towards the number in parentheses and not to the entire expression <math>c\left(\frac{1}{2}\right)</math>. You may also note that <math>\left(\frac{1}{2}\right)^2-3\left(\frac{1}{2}\right)</math> is another way of writing the output. |
*Remind students how to evaluate a function at a given input. | *Remind students how to evaluate a function at a given input. | ||
Latest revision as of 12:17, 1 September 2020
Contents
Objectives:[edit]
- Interpret inputs and outputs of a function
- Evaluate a function at a given input and solve a function equation with a given output
- Extend objectives (i) and (ii) to graphs of functions
- Definitions
- There are no major definitions in this section.
Lesson Guide[edit]
(Warm-Up) Have students do Problems 1 and 2.
Note: Students may not actually remember how to solve 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 H(x)=9}
in Problem 2. Don't get too bogged down on that here.
Interpret inputs and outputs of a function[edit]
Have students spend about 10 minutes working on Problem 3 in their groups. This problem incorporates a function given in words that does not have a formula. You may want to remind your students that this is still a valid function, and you will certainly want to model correct language for them. As you circle the room make sure to interrupt the class and:
- remind students of how to determine what objects are the inputs or outputs of a function,
- remind students that f(input) is an output value,
- emphasize writing the units given in the Problem and a complete sentence, and
- get students comfortable with evaluating a function at a given input.
Use the document camera or the whiteboard and have a group present their answers to Problem 3.
Some questions to ask students in your discussion include: 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 f(103)} a function or a number? What does 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(15)=73} mean on a graph? Would we expect this function to be increasing or decreasing? (You may want to mention to your students that we will ask them to interpret functions in complete sentences on exams.)
Evaluate a function at a given input and solve a function equation with a given output[edit]
Often we are given formulas for functions and asked to evaluate at a given input or solve for a given output. Do an example like Problem 4 to illustrate the difference between evaluating and solving.
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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 |
Note:Contrast language here with previous example. That is emphasize that we're using words like ``find" and ``solve". Make sure your input arrows point towards the number in parentheses and not to the entire expression 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 c\left(\frac{1}{2}\right)} . You may also 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 \left(\frac{1}{2}\right)^2-3\left(\frac{1}{2}\right)} is another way of writing the output.
- Remind students how to evaluate a function at a given input.
- Given an output 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} , emphasize to students that solving 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 f(x)=y} is not the same as plugging the value 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} into the function as an input.
Have the students work on Problem 4
Have students do Problem 5.
Extend objectives (i) and (ii) to graphs of functions[edit]
Have students do Problem 6.
In part (a) students may struggle with the idea that the output is the velocity. Note in part (b) that we don't know for sure, but it's probably 8 minutes, as he stops suddenly and then walks back the way he came. On part (c), students may be tempted to say intervals where he is walking at 3mph, but his fastest speed is actually 4mph. On (d), students may give points rather than intervals for their answers (e.g. "t=1 and t=4" instead of "[1,4]" or "1<t<4")
Once students have worked for a while lead a whole class discussion on the problem.
Have students do Problem 7 and 8. If time permits you should have a group present their answer for Problem 8
Comments[edit]
Some hw problems involve solving for equations like (x+1)/(x+3), and composing a funciton g(x) with 1/(x+2). I recommend doing an example like these so they have them for reference. (This will not be an issue for Fall 2019 onward.)
