1.1 Functions
Objectives:[edit]
Students will be able to:
- Identify whether a given relation is a function
- Identify the inputs and the outputs of a given relation/function
- Describe a given relation/function using multiple representations (tables, ordered pairs, graphs, equations, etc.)
- Review graphs and the vertical line test
- Definitions
- function
Lesson Guide[edit]
Determine when a relationship is a function and determine its inputs and outputs[edit]
[15 minutes] Begin this lesson by giving examples of relationships between two sets. Examples should be easy to understand. I.e., use real world examples such as the days of the year and their average daily temperatures or the students in your class and the month they were born. Discuss how we can identify objects in the two sets and call this identification a “relation.”
A lot of this class will be about exploring the relationship between different pieces of information. For example:
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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 |
If we have a very special kind of relationship called a ``function," we can use one piece of information, called the input, to completely predict the other piece of information, called the output. It is often useful to take a few minutes and draw bubble diagrams of the examples you are going over. E.g.
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You'll now want to define what a function is. Start by using the terms input and output to describe bubbles in your bubble diagram. Say the definition twice for emphasis.
A function is a relation (or a rule) that assigns each input to only one output.
We are not introducing function notation just yet, that comes shortly. Stress the importance of using and understanding precise language. Tell your students that if they are ever asked if a relationship is a function, they have to show that the relation either fulfills or violates the definition of a function.
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Instructor Example |
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This is a sentence that is in place to generate space in the table |
Is the number of letters a function of a word? First, identify what is the input and what is the output. For example:
In this case, the number of letters is a function of the word, since for each input (word) has exactly one output (number of letters). Students may notice that there are some words that have multiple accepted spellings (e.g. color and colour). Let the discussion progress naturally; have them consider whether it is still a function. For example, you may institute the rule that words with different spellings are different words, or else agree to some master list of the proper way to spell each word. Use this conversation to emphasize being precise and careful with language. |
Take one of your previous examples of a function and identify the inputs and outputs. At this point you should also give a relationship that is a non-example of a function and show why it violates the definition of a function.
[10 minutes] Have students do Problems 1-3.
[10 minutes] Ask students to volunteer their answers for Problems 1 and 3 and introduce function notation using these examples.
Note: Students will have difficulty in the preceding question determining which one is the input and which is the output. Explain that when it says "function of (blank)" that this means that (blank) is the input. This concept can also be explained in tandem with function notation below.
Introduce Function Notation[edit]
When we do have a function, we use a mathematical shortcut to talk about it. We write output = f(input), and we can use any letter to represent the function, the input, or the output. Write one of the real-world examples you discussed in class using function notation. For example, to indicate that a quantity y is a function of quantity x, we write y = f(x) and say "y equals f of x". Note: f(x) represents the output of the function f, when x is the input.
To indicate that the output is a function of the input, we express functions as output = f(input). So y = f(x) means x is the input of the function and y is the output of the function. We verbally express this as “y equals f of x.”
It helps to draw the function machine for them to really associate that f(x) is the output.
Remind students that we can use any letter to represent the function, the input, or the output. Also, emphasize that f(x) also represents the output of the function f when x is the input.
Represent relations/functions as tables, ordered pairs, graphs, equations, etc.[edit]
[5 minutes] Choose an example of a function to model in a table, as ordered pairs, and as an equation. Students will encounter functions in all of these forms, and thus should be familiar with all of them. Make sure to plan this example before class so that you can look for an example that works well in all three cases.
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This is a sentence that is in place to generate space in the table |
Words: For each gallon of paint, a painter can paint 250 ft$^2$ of wall. Table: Note that this table is not complete (e.g. what about n=1.5?), but we can infer what f(1.5) is.
Ordered Pairs: {(0,0), (1, 250), (2,500), (3, 750), (4, 1000), (5, 1250), (6, 1500)} Equation/Formula : 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(n) = 250n}
Remind students how to graph functions (the inputs go along the x-axis and the outputs go along the y-axis, etc.). Tell them that this is a fourth major way to represent functions. |
[5 minutes] Have students do Problems 4-5.
Review graphs and the vertical line test[edit]
[5 minutes] Now, draw several graphs and ask how one would know whether or not it represents a function. Students will most likely be reminded of the Vertical Line Test, but it is important in lecture to connect the Vertical Line Test with the formal definition of a function. Be sure to include a graph which is not a function.
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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 |
If a graph fails the vertical line test it cannot be the graph of a function. |
[10 minutes] Have students do Problems 6-9.
[5 minutes] Have students do the synthesis problem.
[10 minutes] Finish class by having groups of students present their answer to the Synthesis Problem using the document camera. Take a few minutes at the end to use the synthesis problem and try to tie everything together.
Comments[edit]
When giving examples of graphs to test the vertical line test on, consider giving a function that is not defined at every real number. Since we don't talk about domain as a part of the definition for a function, students may assume that piecewise functions with breaks can't be functions because they aren't defined at every real number. The vertical line test can give this impression.
If you are giving them time for the survey on this day be careful about how you divide up your time.
Individual Lesson Plans[edit]
Brummer 1.1 Function Plan - For coordinated course
