Lesson 1.5

Range Identification through Graphical Analysis

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

1

Introduction

Prerequisite Connection: 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 Increment: 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.

Why This Matters for Calculus: 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.

2

Explanation of 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)?
3

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.

4

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.

5

Real-World Application

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.

6

Practice Quiz

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