# Lesson 5

Using Function Notation to Describe Rules (Part 2)

### Lesson Narrative

In an earlier lesson, students learned that some functions can be defined with a rule and the rule can be expressed using function notation. In this lesson, students use rules of functions to find the output when the input is given (or to evaluate functions) and to find the input when the output is known (or to solve equations that define functions). They also interpret rules of functions in terms of a situation. Along the way, they practice reasoning quantitatively and abstractly (MP2).

The term linear function is introduced here. In middle school, students learned a relationship between two quantities is linear if one quantity changes at a constant rate relative to the other. Students see that a linear function can be understood in similar terms: a function is linear if the output changes by a constant rate relative to its input.

This lesson includes an optional activity that is designed to enable students to use technology to graph and evaluate functions expressed in function notation. This skill can help to develop students’ understanding of functions and ability to solve problems in this unit and in future units.

### Learning Goals

Teacher Facing

• Evaluate functions and solve equations given in function notation, either by graphing or by reasoning algebraically.
• Understand linear function as a function whose output changes at a constant rate and whose graph is a line.
• Use technology to graph and evaluate functions given in function notation.

### Student Facing

Let’s graph and find the values of some functions.

### Required Preparation

For the optional activity Function Notation and Graphing Technology, acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

### Student Facing

• I can use technology to graph a function given in function notation, and use the graph to find the values of the function.
• I know different ways to find the value of a function and to solve equations written in function notation.
• I know what makes a function a linear function.

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.

• linear function

A linear function is a function that has a constant rate of change. Another way to say this is that it grows by equal differences over equal intervals. For example, $$f(x)=4x-3$$ defines a linear function. Any time $$x$$ increases by 1, $$f(x)$$ increases by 4.