# Lesson 4

Using Function Notation to Describe Rules (Part 1)

### Lesson Narrative

In earlier lessons, students interpreted and wrote statements in function notation to represent specific input-output pairs of a function (such as $$p(2.5)=18$$) or relationships between specific pairs (such as $$W(10)=W(12)$$).

In this lesson, students learn that function notation can also be used to describe the rule of a function or how a function behaves generally, at any value of input. For instance, they see that if the output of a function $$f$$ can be found by multiplying the input by 3 and then subtracting 10 from the result, we can write $$f(x) = 3x - 10$$ to represent this rule. We can also use this rule (either the verbal description or the equation) to find the output for any input. In some cases, the rule can also be used to find the input when we know the output.

Students continue to decontextualize given situations into symbolic representations and to contextualize the latter in order to solve problems (MP2). To connect different representations of functions defined by rules, they look for and make use of structure (MP7).

### Learning Goals

Teacher Facing

• Create tables and graphs to represent a function given statements in function notation.
• Interpret rules of functions that are expressed using function notation.
• Use function notation to write equations that represent rules of functions.

### Student Facing

Let’s look at some rules that describe functions and write some, too.

### Student Facing

• I can make sense of rules of functions when they are written in function notation, and create tables and graphs to represent the functions.
• I can write equations that represent the rules of functions.

Building On

Building Towards

### Glossary Entries

• dependent variable

A variable representing the output of a function.

The equation $$y = 6-x$$ defines $$y$$ as a function of $$x$$. The variable $$x$$ is the independent variable, because you can choose any value for it. The variable $$y$$ is called the dependent variable, because it depends on $$x$$. Once you have chosen a value for $$x$$, the value of $$y$$ is determined.

• function

A function takes inputs from one set and assigns them to outputs from another set, assigning exactly one output to each input.

• function notation

Function notation is a way of writing the outputs of a function that you have given a name to. If the function is named $$f$$ and $$x$$ is an input, then $$f(x)$$ denotes the corresponding output.

• independent variable

A variable representing the input of a function.

The equation $$y = 6-x$$ defines $$y$$ as a function of $$x$$. The variable $$x$$ is the independent variable, because you can choose any value for it. The variable $$y$$ is called the dependent variable, because it depends on $$x$$. Once you have chosen a value for $$x$$, the value of $$y$$ is determined.