1.7 Domain & Range

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Objectives:

• Identify the domain and range of a function represented in various forms

Important Items

Definitions:

domain, range

Notes:

We often will model discrete situations using continuous functions. We will {\em not} be covering finding the domain of a discrete situation (although there are some examples of this in the text). Tell students that unless a problem asks for a specific format of notation, it is acceptable to use whichever is most familiar, i.e., the domain of 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)=\sqrt{x}} is any of: 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\leq x} , 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:x\geq 0\}} , or 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, \infty)} .

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Lesson Guide

Warm-Up

Have students do Problems 1 and 2.

Have students work on problem 3. You can use this problem to lead into your discussion of domain and range.

Identify the domain and range of a function represented in various forms

If 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=f(x)}
 is a function, then
* The domain of 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}
 is the set of input values, 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}
, which yield an output value. 
* The range of 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}
 is the corresponding set of output values, 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}
.

Give students multiple ways to think about the domain and range; i.e., the domain is the ``set of allowed inputs. Additionally, you may graph a function in black and use a different color to trace along the 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} -axis to denote the domain and yet another color to trace along the 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} -axis to denote the range.

Have students do Problem 4.

Do several examples similar to Problems 4-6. Examples should show different ways in which one can determine the domain and range (use a table, ordered pairs, a graph, a formula, and a word problem). Also be sure to give an example of when something is NOT in the domain or NOT in the range of a given function. It might be helpful to use bubble diagrams of the functions here. e.g.

Instructor Example	Suggested Example
This is a sentence that is in place to generate space in the table	Find the domain of 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)=\sqrt{x+8}} .

Instructor Example

Suggested Example

This is a sentence that is in place to generate space in the table

Find the range of 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)=\sqrt{x+8}} . From the graph we can see the the range 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 ([0,\infty)} .

Have students do Problems 5-11. This section is consistently difficult for students. Throughout class, asking students to present their solutions on the board and also giving more examples as questions arise will be helpful.

You should also do an example of how to find the range of a function if you specify a domain for a function, because this appears a lot on the homework. Something linear may work nicely.

Comments

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• Problem 9 should reference Problem 4 rather than Problem 3.
• The spacing on Problem 3 should be fixed for the next version of the course packet (misplaced \newpage command)