Experience First

Although students have used the words domain and range for many years, this lesson focuses on determining domain and range in multiple representations. Students are asked to connect graphical representations to numerical representations and algebraic representations, noticing where a given x-value has no ordered pair associated with it on the graph, produces an ERROR when evaluating a function on the calculator, or causes a function to be undefined. 

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When monitoring groups, ask questions that have students articulate why the domain of a given function is restricted to certain values. Students should be able to communicate about vertical asymptotes in the case of rational functions, and how square rooting a negative number yields imaginary values which are not seen on a graph.

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Formalize Later

This lesson focuses on square root functions (particularly those that represent semicircles) and rational functions. Students often have a hard time finding the domain analytically because they tend to rely on memorization and tricks instead of connecting the functions to their graphs. Students often confuse how to find the domain of functions like sqrt(x^2-49) and the sqrt (49-x^2). Have students see the difference by looking at the graphs but also solving algebraically. This can be scaffolded in the following way:

• Can I plug in 6? Why or why not? Can I plug in 7? Why or why not? Can I plug in 8? What about 7.1?

• Why do some values work but others don’t? 

• What’s the tipping point? (i.e. When do we start getting “errors”?)

• What do you know about square roots? What does that tell us about the expression under the square root?

• What does x^2 have to be so that x^2-49 is positive? What values of x would make this happen?

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Invite students to test particular values before noticing patterns and making generalizations. There are two common misconceptions about domain. The first is the false idea that the square root of 0 is undefined. Ask students if they can think of a number that when multiplied by itself gives 0. They should conclude that 0 satisfies this condition! 

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The second misconception comes up more frequently with rational functions and that is that if numbers less than or equal to 7 are not allowed, then numbers that are greater than or equal to 8 are allowed. They forget about all the values between 7 and 8! In your questioning, ask students about some decimal values to drive home the idea that, in the above example, any number bigger than 7 would be allowed, even 7.00001. This reinforces the idea of interval notation and the use of a parenthesis. 

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The first problem in the Check Your Understanding helps students think about domain and range contextually, and explore the difference between the inputs for which the function makes sense in a given context and the inputs for which the algebraic model of a scenario has a defined value. This is a good opportunity to talk about the limitations of models and their purpose in mathematics.