Lesson 4.5

Domain and Range (Continuous)

When the points connect, lists aren't enough. We need intervals. Scan across the horizon, then scan up the sky.

Introduction

In Lesson 4.4, we listed specific points (Discrete). But most graphs are smooth lines or curves (Continuous). We can't list every number between 1 and 2 (1.1, 1.01, 1.001...), so we use Inequalities or Interval Notation.

Past Knowledge

Lesson 3.3 (Inequalities). Remember open circles () vs closed circles ().

Today's Goal

Find Domain (Left to Right) and Range (Bottom to Top) of continuous graphs.

Future Success

This skill is vital for analyzing polynomial functions and limits in Calculus.

Key Concepts

The Trash Compactor

Imagine squishing the graph flat against the axes.

Finding Domain

Scan Left → Right

What is the furthest left x-value? What is the furthest right x-value?

Finding Range

Scan Bottom ↑ Top

What is the lowest y-value? What is the highest y-value?

Worked Examples

Example 1: A Line Segment

Basic

Find Domain and Range.

Domain (Left/Right)

Reviewing X-axis: Starts at -2 (Closed), Ends at 3 (Closed).

Range (Bottom/Top)

Reviewing Y-axis: Lowest point is -1. Highest point is 4.

Example 2: Open Circles

Intermediate

Find Domain and Range.

Domain

Left: -3 (Open Circle → Just $<$). Right: 2 (Closed Circle → $\le$).

Range

Low: -2 (Closed). High: 3 (Open).

Example 3: Infinity and Beyond

Advanced

Find Domain and Range.

Domain

It goes left forever and right forever. All Real Numbers.

or

Range

The lowest point is $y = 0$. It goes up forever.

Common Pitfalls

Scanning Wrong Direction

Always scan Left-to-Right (Small-to-Big). Never say . It must be .

The Vertical Line Trap

Students confuse the Vertical Line Test (for functions) with Domain. For Domain, you are looking at the SHADOW on the x-axis.

Real-Life Applications

Roller Coasters: The track is a continuous function. The Domain is the time from start to finish (0 to 120 seconds). The Range is the height (Ground level to Peak height). Knowing the range tells engineers how high the supports need to be.

Practice Quiz

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