Lesson 1.5

Range Identification through Graphical Analysis

Algebra tells us what inputs are allowed. Graphs tell us what outputs are possible.

Introduction

Algebra tells us what inputs are allowed. Graphs tell us what outputs are possible.

Past Knowledge

You already know how to find the Domain by scanning a graph from left-to-right (x-axis). Finding the Range is the exact same skill, but rotated 90 degrees.

Today's Goal

We define the Range as the set of all possible output values (-coordinates). You will learn to scan graphs from Bottom-to-Top ("Floor to Ceiling") to identify these values.

Future Success

In Chapter 3, you will learn about Inverse Functions. The Range of a function becomes the Domain of its inverse. If you can't find the range now, you won't be able to define the inverse later.

Key Concepts

The "Floor to Ceiling" Scan

Scanning Up...
  • 1
    Start at the Bottom: Look at the lowest point on the graph. Does it stop at a number, or does it have an arrow pointing down to ?
  • 2
    Move Upwards: Scan up the y-axis. Are there any gaps, breaks, or horizontal asymptotes?
  • 3
    Check the Ceiling: Look at the highest point. Does it stop at a peak (bracket), or go up forever (infinity)?

Worked Examples

Level: Basic

Example 1: The Parabola

Find the range of .

Step 1: Floor Check
The lowest y-value is 0. The graph touches zero, so we use a bracket: .
Step 2: Ceiling Check
The arrows point up forever. There is no ceiling.
Answer
Level: Intermediate

Example 2: Shifted Function

Find the range of .

Step 1: Floor Check
The lowest point is y = -3. It touches, so: .
Answer
Level: Advanced (Calculus Prep)

Example 3: Horizontal Asymptote

Find the range of .

The Scan
The graph goes from all the way up to... 2. Then it skips 2. Then it continues from immediately above 2 to .
Answer

Calculus Insight: The value 2 is the "limit at infinity." The function approaches it but never reaches it.

Common Pitfalls

  • Scanning Top-to-Bottom:

    If you read top-down, you will write your interval backwards, like . Intervals MUST be written Smallest to Largest (Left to Right on the number line).

  • Confusing Axes:

    Domain is x (left/right). Range is y (down/up). If you mix them up, you are describing the wrong set of numbers.

Real-Life Applications

Engineering: Stress Tolerances

Consider a function that represents the stress on a bridge beam over time.

The Range of this function tells engineers the absolute minimum and maximum stress the beam will ever experience.

Range = [Min Stress, Max Stress]

If the maximum stress (the ceiling of the range) exceeds the material's breaking point, the bridge collapses. Finding the range is literally a matter of safety.

Practice Quiz

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