Understanding Functions Visually: Key Concepts from Stewart’s Calculus

Intro:

I’ve started reading Calculus: Early Transcendentals by James Stewart as part of a deeper dive into math fundamentals. In this post, I’m sharing my notes and visual examples from Chapter 1 — specifically focusing on how we think about functions as rules, graphs, and real-world models.

What This Covers:

What is a function?

Domain and range

Graph transformations (shifts, flips, stretches)

Piecewise functions

Inverse functions (at a glance)

1. What is a Function?

A function is a rule that assigns each input exactly one output.
Example:
Let’s say f(x) = x². This means:

f(2) = 4

f(-2) = 4

But not: one input → two outputs ❌

2. Domain and Range

The domain is all inputs where the function makes sense.
The range is all outputs that result.

f(x) = √(x - 2)
Domain: x ≥ 2
Range: y ≥ 0

3. Transforming Graphs

f(x - 2) shifts right

-f(x) reflects over x-axis

f(2x) compresses horizontally

Visual example:

“If f(x) = x², then g(x) = (x - 2)² is f shifted 2 units right.”

4. Piecewise Functions

Example:

f(x) = {
x + 1, if x < 0
x², if x ≥ 0
}

5. Inverse Functions (Intro)

Inverses “undo” the function: If f(x) = 2x + 3, then f⁻¹(x) = (x - 3)/2

• Reflection over y = x

• Not all functions have inverses (unless 1-to-1)

Final Thoughts

• Chapter 1 lays the groundwork for all calculus

• I’m focusing on visual understanding + modeling

• Future posts will explore limits and continuity

Tags:

#calculus #math #students #C++ #beginner #visuallearning