Nebraska Open Access Mathematics - User contributions [en]

1.6 Function Notation Input & Output

2020-09-01T12:17:07Z

Kkelly12: /* Evaluate a function at a given input and solve a function equation with a given output */

[[1.5 Comparing Linear Functions | Prior Lesson]] | [[1.7 Domain & Range | Next Lesson]]

==Objectives:==

*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==

(Warm-Up) Have students do Problems 1 and 2.

Note: Students may not actually remember how to solve <math>H(x)=9</math> in Problem 2. Don't get too bogged down on that here.

===Interpret inputs and outputs of a function===

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 <math>f(103)</math> a function or a number? What does <math>f(15)=73</math> 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===

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.

{{Ex-begin}}
[[File:Screen Shot 2019-12-05 at 3.53.05 PM.png|thumb]]
{{Ex-end}}

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.
*Given an output <math>y</math>, emphasize to students that solving the equation <math>f(x)=y</math> is not the same as plugging the value <math>y</math> 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===

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==
----
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.)

1.6 Function Notation Input & Output

2020-09-01T12:16:51Z

Kkelly12: /* Evaluate a function at a given input and solve a function equation with a given output */

[[1.5 Comparing Linear Functions | Prior Lesson]] | [[1.7 Domain & Range | Next Lesson]]

==Objectives:==

*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==

(Warm-Up) Have students do Problems 1 and 2.

Note: Students may not actually remember how to solve <math>H(x)=9</math> in Problem 2. Don't get too bogged down on that here.

===Interpret inputs and outputs of a function===

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 <math>f(103)</math> a function or a number? What does <math>f(15)=73</math> 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===

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.

{{Ex-begin}}
[[File:Screen Shot 2019-12-05 at 3.53.05 PM.png|thumb]]
{{Ex-end}}

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.
*Given an output <math>y</math>, emphasize to students that solving the equation f(x)=y is not the same as plugging the value <math>y</math> 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===

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==
----
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.)

1.6 Function Notation Input & Output

2020-09-01T12:15:53Z

Kkelly12: /* Evaluate a function at a given input and solve a function equation with a given output */

[[1.5 Comparing Linear Functions | Prior Lesson]] | [[1.7 Domain & Range | Next Lesson]]

==Objectives:==

*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==

(Warm-Up) Have students do Problems 1 and 2.

Note: Students may not actually remember how to solve <math>H(x)=9</math> in Problem 2. Don't get too bogged down on that here.

===Interpret inputs and outputs of a function===

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 <math>f(103)</math> a function or a number? What does <math>f(15)=73</math> 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===

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.

{{Ex-begin}}
[[File:Screen Shot 2019-12-05 at 3.53.05 PM.png|thumb]]
{{Ex-end}}

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>(\frac{1}{2})^2-3(\frac{1}{2})</math> is another way of writing the output.

*Remind students how to evaluate a function at a given input.
*Given an output <math>y</math>, emphasize to students that solving the equation <math>f(x)=y</math> is not the same as plugging the value <math>y</math> 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===

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==
----
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.)

1.1 Functions

2020-07-17T16:54:04Z

Kkelly12: /* Objectives: */

[[Covariational Reasoning | Prior Lesson]] | [[1.2 Rate of Change | Next Lesson]]

==Objectives:==
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==

====Determine when a relationship is a function and determine its inputs and outputs====
[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:

{{Ex-begin}}

*The year you were born is related to your age.
*The month is related to the temperature outside.
*The price of gold is related to the supply and the demand.

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.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/isrsxtlsgg?embed
|width=500
|height=500
|border=0
}}
{{Ex-end}}

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.

{{Ex-begin}}
Is the number of letters a function of a word?

First, identify what is the input and what is the output. For example:

{| class="wikitable"
! Input: Word
! Output: Number of Letters
|-
| I
| 1
|-
| Love
| 4
|-
| Math
| 4
|-
| Class
| 5
|}
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.

{{Ex-end}}

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====

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.====
[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.

{{Ex-begin}}

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.

{| class="wikitable"
! n
! 0
! 1
! 2
! 3
! 4
! 5
! 6
|-
| f(n)
| 0
| 250
| 500
| 750
| 1000
| 1250
| 1500
|}

Ordered Pairs: {(0,0), (1, 250), (2,500), (3, 750), (4, 1000), (5, 1250), (6, 1500)}

Equation/Formula : <math> f(n) = 250n</math>

Graph:
[[File:Paintinggraph.png|thumb]]

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.
{{Ex-end}}

[5 minutes] Have students do Problems 4-5.

====Review graphs and the vertical line test====

[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.

{{Ex-begin}}
If a graph fails the vertical line test it cannot be the graph of a function.

[[File:VLT2.png|thumb]]

{{Ex-end}}

[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==
----
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==

[[Brummer 1.1 Function Plan]] - For coordinated course

1.1 Functions

2020-07-17T16:46:00Z

Kkelly12: /* Objectives: */

[[Covariational Reasoning | Prior Lesson]] | [[1.2 Rate of Change | Next Lesson]]

==Objectives:==
Students will be able to:
*Identify whether a relationship is a function
*Identify the inputs and the outputs of a function
*Represent relations/functions as tables, ordered pairs, graphs, equations, etc.
*Review graphs and the vertical line test

;Definitions: function

----

==Lesson Guide==

====Determine when a relationship is a function and determine its inputs and outputs====
[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:

{{Ex-begin}}

*The year you were born is related to your age.
*The month is related to the temperature outside.
*The price of gold is related to the supply and the demand.

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.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/isrsxtlsgg?embed
|width=500
|height=500
|border=0
}}
{{Ex-end}}

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.

{{Ex-begin}}
Is the number of letters a function of a word?

First, identify what is the input and what is the output. For example:

{| class="wikitable"
! Input: Word
! Output: Number of Letters
|-
| I
| 1
|-
| Love
| 4
|-
| Math
| 4
|-
| Class
| 5
|}
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.

{{Ex-end}}

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====

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.====
[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.

{{Ex-begin}}

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.

{| class="wikitable"
! n
! 0
! 1
! 2
! 3
! 4
! 5
! 6
|-
| f(n)
| 0
| 250
| 500
| 750
| 1000
| 1250
| 1500
|}

Ordered Pairs: {(0,0), (1, 250), (2,500), (3, 750), (4, 1000), (5, 1250), (6, 1500)}

Equation/Formula : <math> f(n) = 250n</math>

Graph:
[[File:Paintinggraph.png|thumb]]

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.
{{Ex-end}}

[5 minutes] Have students do Problems 4-5.

====Review graphs and the vertical line test====

[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.

{{Ex-begin}}
If a graph fails the vertical line test it cannot be the graph of a function.

[[File:VLT2.png|thumb]]

{{Ex-end}}

[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==
----
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==

[[Brummer 1.1 Function Plan]] - For coordinated course