Function Notation

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Lessons

Standards in this Lesson

Common Core Math Standards

HSF.BF.A: Build a function that models a relationship between two quantities.
HSF.BF.A.1: Write a function that describes a relationship between two quantities.
HSF.IF.A: Understand the concept of a function and use function notation.
HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Alabama Standards

AL.HSF.BF.A: Build a function that models a relationship between two quantities.
AL.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
AL.HSF.IF.A: Understand the concept of a function and use function notation.
AL.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
AL.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Arkansas Standards

AR.HSF.BF.A: Build a function that models a relationship between two quantities.
AR.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
AR.HSF.IF.A: Understand the concept of a function and use function notation.
AR.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
AR.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

California Standards

CA.HSF.BF.A: Build a function that models a relationship between two quantities.
CA.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
CA.HSF.IF.A: Understand the concept of a function and use function notation.
CA.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
CA.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Colorado Standards

CO.HSF.BF.A: Build a function that models a relationship between two quantities.
CO.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
CO.HSF.IF.A: Understand the concept of a function and use function notation.
CO.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
CO.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Connecticut Standards

CT.HSF.BF.A: Build a function that models a relationship between two quantities.
CT.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
CT.HSF.IF.A: Understand the concept of a function and use function notation.
CT.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
CT.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Delaware Standards

DE.HSF.BF.A: Build a function that models a relationship between two quantities.
DE.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
DE.HSF.IF.A: Understand the concept of a function and use function notation.
DE.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
DE.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Delaware Standards

GA.HSF.BF.A: Build a function that models a relationship between two quantities.
GA.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
GA.HSF.IF.A: Understand the concept of a function and use function notation.
GA.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
GA.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Hawaii Standards

HI.HSF.BF.A: Build a function that models a relationship between two quantities.
HI.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
HI.HSF.IF.A: Understand the concept of a function and use function notation.
HI.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
HI.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Idaho Standards

ID.HSF.BF.A: Build a function that models a relationship between two quantities.
ID.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
ID.HSF.IF.A: Understand the concept of a function and use function notation.
ID.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
ID.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Illinois Standards

IL.HSF.BF.A: Build a function that models a relationship between two quantities.
IL.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
IL.HSF.IF.A: Understand the concept of a function and use function notation.
IL.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
IL.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Iowa Standards

IA.HSF.BF.A: Build a function that models a relationship between two quantities.
IA.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
IA.HSF.IF.A: Understand the concept of a function and use function notation.
IA.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
IA.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Kansas Standards

KS.HSF.BF.A: Build a function that models a relationship between two quantities.
KS.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
KS.HSF.IF.A: Understand the concept of a function and use function notation.
KS.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
KS.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Kentucky Standards

KY.HSF.BF.A: Build a function that models a relationship between two quantities.
KY.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
KY.HSF.IF.A: Understand the concept of a function and use function notation.
KY.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
KY.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Louisiana Standards

LA.HSF.BF.A: Build a function that models a relationship between two quantities.
LA.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
LA.HSF.IF.A: Understand the concept of a function and use function notation.
LA.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
LA.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Maine Standards

ME.HSF.BF.A: Build a function that models a relationship between two quantities.
ME.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
ME.HSF.IF.A: Understand the concept of a function and use function notation.
ME.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
ME.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Michigan Standards

MI.HSF.BF.A: Build a function that models a relationship between two quantities.
MI.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
MI.HSF.IF.A: Understand the concept of a function and use function notation.
MI.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
MI.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Maryland Standards

MD.HSF.BF.A: Build a function that models a relationship between two quantities.
MD.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
MD.HSF.IF.A: Understand the concept of a function and use function notation.
MD.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
MD.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Massachusetts Standards

MA.HSF.BF.A: Build a function that models a relationship between two quantities.
MA.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
MA.HSF.IF.A: Understand the concept of a function and use function notation.
MA.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
MA.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Mississippi Standards

MS.HSF.BF.A: Build a function that models a relationship between two quantities.
MS.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
MS.HSF.IF.A: Understand the concept of a function and use function notation.
MS.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
MS.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Missouri Standards

MO.HSF.BF.A: Build a function that models a relationship between two quantities.
MO.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
MO.HSF.IF.A: Understand the concept of a function and use function notation.
MO.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
MO.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Montana Standards

MT.HSF.BF.A: Build a function that models a relationship between two quantities.
MT.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
MT.HSF.IF.A: Understand the concept of a function and use function notation.
MT.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
MT.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

New Hampshire Standards

NH.HSF.BF.A: Build a function that models a relationship between two quantities.
NH.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
NH.HSF.IF.A: Understand the concept of a function and use function notation.
NH.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
NH.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Nevada Standards

NV.HSF.BF.A: Build a function that models a relationship between two quantities.
NV.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
NV.HSF.IF.A: Understand the concept of a function and use function notation.
NV.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
NV.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

New Jersey Standards

NJ.HSF.BF.A: Build a function that models a relationship between two quantities.
NJ.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
NJ.HSF.IF.A: Understand the concept of a function and use function notation.
NJ.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
NJ.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

New Mexico Standards

NM.HSF.BF.A: Build a function that models a relationship between two quantities.
NM.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
NM.HSF.IF.A: Understand the concept of a function and use function notation.
NM.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
NM.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

North Carolina Standards

NC.HSF.BF.A: Build a function that models a relationship between two quantities.
NC.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
NC.HSF.IF.A: Understand the concept of a function and use function notation.
NC.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
NC.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

North Dakota Standards

ND.HSF.BF.A: Build a function that models a relationship between two quantities.
ND.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
ND.HSF.IF.A: Understand the concept of a function and use function notation.
ND.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
ND.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

New York Standards

NY.HSF.BF.A: Build a function that models a relationship between two quantities.
NY.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
NY.HSF.IF.A: Understand the concept of a function and use function notation.
NY.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
NY.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Ohio Standards

OH.HSF.BF.A: Build a function that models a relationship between two quantities.
OH.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
OH.HSF.IF.A: Understand the concept of a function and use function notation.
OH.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
OH.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Oregon Standards

OR.HSF.BF.A: Build a function that models a relationship between two quantities.
OR.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
OR.HSF.IF.A: Understand the concept of a function and use function notation.
OR.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
OR.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Pennsylvania Standards

PA.HSF.BF.A: Build a function that models a relationship between two quantities.
PA.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
PA.HSF.IF.A: Understand the concept of a function and use function notation.
PA.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
PA.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Rhode Island Standards

RI.HSF.BF.A: Build a function that models a relationship between two quantities.
RI.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
RI.HSF.IF.A: Understand the concept of a function and use function notation.
RI.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
RI.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

South Dakota Standards

SD.HSF.BF.A: Build a function that models a relationship between two quantities.
SD.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
SD.HSF.IF.A: Understand the concept of a function and use function notation.
SD.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
SD.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Tennessee Standards

TN.HSF.BF.A: Build a function that models a relationship between two quantities.
TN.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
TN.HSF.IF.A: Understand the concept of a function and use function notation.
TN.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
TN.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Utah Standards

UT.HSF.BF.A: Build a function that models a relationship between two quantities.
UT.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
UT.HSF.IF.A: Understand the concept of a function and use function notation.
UT.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
UT.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Vermont Standards

VT.HSF.BF.A: Build a function that models a relationship between two quantities.
VT.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
VT.HSF.IF.A: Understand the concept of a function and use function notation.
VT.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
VT.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Washington Standards

WA.HSF.BF.A: Build a function that models a relationship between two quantities.
WA.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
WA.HSF.IF.A: Understand the concept of a function and use function notation.
WA.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
WA.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

West Virginia Standards

WV.HSF.BF.A: Build a function that models a relationship between two quantities.
WV.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
WV.HSF.IF.A: Understand the concept of a function and use function notation.
WV.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
WV.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Wisconsin Standards

WI.HSF.BF.A: Build a function that models a relationship between two quantities.
WI.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
WI.HSF.IF.A: Understand the concept of a function and use function notation.
WI.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
WI.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Wyoming Standards

WY.HSF.BF.A: Build a function that models a relationship between two quantities.
WY.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
WY.HSF.IF.A: Understand the concept of a function and use function notation.
WY.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
WY.HSF.IF.B: Interpret functions that arise in applications in terms of the context.

Practices in this Lesson

Math

MP.4: Model with mathematics
MP.6: Attend to precision
MP.7: Look for and make use of structure

(Also available in Pyret)

Students learn to read function notation and evaluate expressions using function definitions, tables, and graphs.

Lesson Goals

Students will be able to:

Identify the value that applying a function (e.g. - f(3)) produces from symbolic definitions, tables, or graphs.

Student-Facing Lesson Goals

I can find the value of an expression such as f(3), given a table, graph or symbolic definiton.

Materials

This lesson is unplugged and does not require a computer.

Glossary

apply: use a given function on some inputs
coordinate plane: a grid formed by a pair of intersecting horizontal and vertical number lines
evaluate: perform the computation in an expression, producing an answer
function: a relation from a set of inputs to a set of possible outputs, where each input is related to exactly one output
function notation: a standard way to denote functions, beginning with a function name, such as f or gt, the variable(s) the function takes in, such as x or size, and a definition of what the function does using a formula in terms of the variable. For example, f(x) = x + 1 takes in a number and adds 1.
line: an infinite collection of points forming a straight one-dimensional figure having no thickness and extending infinitely in both directions
substitute: replace a variable with a value or expression and evaluate or simplify the resulting expression
variable: a name or symbol that stands for some value or expression, often a value or expression that changes

🔗Function Notation (Definitions) 15 minutes

Overview

Students connect their understanding of function definitions in Wescheme to function definitions in math.

Launch

We’ve seen how functions like gt can be defined, and then applied to a number to create a green triangle. And once gt is defined, we can use it with many different numbers to create many different triangles - all without having to write out "solid", "green", etc.

But how does this function work?

When we apply a function to some inputs, we substitute those inputs for the variables in the definition. In the example below, the inputs are substituted for the variable size in the body of the gt function.

(define (gt size) (triangle size "solid" "green"))

Apply the Function Substitute the Input(s) Compute the Answer

Apply the Function	Substitute the Input(s)	Compute the Answer
`(gt 10)`	`(triangle 10 "solid" "green")`	🖼Show image
`(gt 20)`	`(triangle 20 "solid" "green")`	🖼Show image
`(gt 30)`	`(triangle 30 "solid" "green")`	🖼Show image
`(gt 40)`	`(triangle 40 "solid" "green")`	🖼Show image
`(gt 50)`	`(triangle 50 "solid" "green")`	🖼Show image

(gt 10)

(triangle 10 "solid" "green")

green triangle 10 🖼Show image

(gt 20)

(triangle 20 "solid" "green")

green triangle 20 🖼Show image

(gt 30)

(triangle 30 "solid" "green")

green triangle 30 🖼Show image

(gt 40)

(triangle 40 "solid" "green")

green triangle 40 🖼Show image

(gt 50)

(triangle 50 "solid" "green")

green triangle 50 🖼Show image

Let’s take a look at a function that works with Numbers:

(define (f x) (+ x 8))

Apply the Function Substitute the Input(s) Compute the Answer

Apply the Function	Substitute the Input(s)	Compute the Answer
$\displaystyle f(10)$	`(+ 10 8)`	18
$\displaystyle f(20)$	`(+ 20 8)`	28
$\displaystyle f(30)$	`(+ 30 8)`	38
$\displaystyle f(40)$	`(+ 40 8)`	48
$\displaystyle f(50)$	`(+ 50 8)`	58

$\displaystyle f(10)$

(+ 10 8)

$\displaystyle f(20)$

(+ 20 8)

$\displaystyle f(30)$

(+ 30 8)

$\displaystyle f(40)$

(+ 40 8)

$\displaystyle f(50)$

(+ 50 8)

Once again, the input is substituted for the variable.

You’ve already seen that Wescheme looks a little different from traditional math, even if it behaves the same way.

Math books use something called Function Notation to define functions. Here’s a side-by-side comparison of the same function, in our programming language and in function notation:

Defining Functions in Wescheme Defining Functions in Function Notation

Defining Functions in Wescheme	Defining Functions in Function Notation
`(define (f x) (+ x 8))`	$\displaystyle f(x) = x + 8$

(define (f x) (+ x 8))

$\displaystyle f(x) = x + 8$

What do these forms have in common?
- Both forms show the name of the function, as well as the names of the variables. They also show what the function does with those variables.
How are these forms different from one another?
- In Pyret, we use a colon instead of an equals sign. In Pyret, we see fun at the beginning and end at the end.

In math - just as in programming - we compute the value of the expression for any specific input by substituting numbers for the variable(s) used in the definition, just as we did with gt.

$\displaystyle f(x) = x + 8$

Apply the Function	Substitute the Input(s)	Compute the Answer
$\displaystyle f(10)$	$\displaystyle 10 + 8$	18
$\displaystyle f(20)$	$\displaystyle 20 + 8$	28
$\displaystyle f(30)$	$\displaystyle 30 + 8$	38
$\displaystyle f(40)$	$\displaystyle 40 + 8$	48
$\displaystyle f(50)$	$\displaystyle 50 + 8$	58

Apply the Function

Substitute the Input(s)

Compute the Answer

$\displaystyle f(10)$

$\displaystyle 10 + 8$

$\displaystyle f(20)$

$\displaystyle 20 + 8$

$\displaystyle f(30)$

$\displaystyle 30 + 8$

$\displaystyle f(40)$

$\displaystyle 40 + 8$

$\displaystyle f(50)$

$\displaystyle 50 + 8$

Investigate

Turn to Matching Examples and Definitions (Math) and Function Notation - Substitution.
Start by looking at each table and highlighting what is changing from the first row to the following rows.
Then, match each table to the function that defines it.

You may also want to have students complete Matching Examples & Function Definitions (Math) (Desmos)

Synthesize

You can think of $\displaystyle f(3)$ as a question.

What question is it asking you to evaluate?
- What is the value of $\displaystyle x + 8$ when $\displaystyle x$ is 3?
What is another way you can ask it?
- What is $\displaystyle 3 + 8$ ?