Foundations of Calculus: Functions, Domains, and Graphs

Section One: Introduction to Calculus

What is Calculus?

Calculus is a branch of mathematics focused on understanding change and accumulation. It deals with finding rates of change (derivatives) and calculating lengths, areas, and volumes (integrals).

• Derivatives: Measure how a quantity changes with respect to another.

• Integrals: Measure accumulation, such as area under a curve.

• Applications: Physics, engineering, economics, biology, and more.

Course Structure

• Grading: 35% of the grade is based on quizzes and homework.

Section Two: Functions and Modules

Definition of a Function

A function is a rule that assigns each element x in a set D to exactly one element called f(x) in a set F.

• Inputs: The set of possible values for x (domain).

• Outputs: The set of possible values for f(x) (range).

Representing Functions

• Functions can be represented by tables, graphs, or mathematical expressions.

The Vertical Line Test

A curve is a function if a vertical line intersects the curve only once at any given x-value.

• If a vertical line crosses a graph more than once, it is not a function.

Piecewise Functions

A piecewise function is defined by different expressions for different intervals of the domain.

• Example: $f(x) = egin{cases} x, & x > 1 \ x^2, & x ext{ otherwise} \\ ext{(stops at } x=1 ext{, but still a function)} \\ ext{Additional info: Piecewise functions are useful for modeling situations with abrupt changes.}$

Even Functions

A function f(x) is even if $f(-x) = f(x)$ for every x in the domain.

• Example: $f(x) = x^2$ is even because $(-x)^2 = x^2$.

Increasing and Decreasing Functions

Functions can be classified by their behavior over intervals:

• Increasing: $f(x)$ increases as $x$ increases.

• Decreasing: $f(x)$ decreases as $x$ increases.

• Monotonic: A function that is always increasing or always decreasing.

Domain and Range

• Domain: The set of all possible input values (x).

• Range: The set of all possible output values (f(x)).

Section Three: Types of Numbers

Classification of Numbers

• Natural numbers: $\mathbb{N}$ (1, 2, 3, ...)

• Integers: $\mathbb{Z}$ (..., -2, -1, 0, 1, 2, ...)

• Rational numbers: $\mathbb{Q}$ (fractions, e.g., 1/2, -3/4)

• Real numbers: $\mathbb{R}$ (all numbers on the number line)

Section Four: Finding Domain and Range

Finding the Domain

• For $f(x) = \sqrt{x}$, domain is $x \geq 0$.

• For $f(x) = \frac{1}{x}$, domain is $x \neq 0$.

• For $f(x) = \sqrt{7 - x}$, domain is $x \leq 7$.

• For $f(x) = \sqrt{x^2 - 9}$, domain is $x \leq -3$ or $x \geq 3$.

Finding the Range

• Graph the function to determine possible output values.

• Example: For $f(x) = \sqrt{x}$, range is $y \geq 0$.

Finding the Missing Piece

• Given $g(x) = x^3$, find $f(x)$ such that $f(g(x)) = x^6$. Since $(x^3)^2 = x^6$, $f(x) = x^2$.

Section Five: Function Types

Algebraic and Transcendental Functions

Algebraic	Transcendental
Polynomial	Trigonometric
Rational	Exponential
Radical	Logarithmic

Section Six: Linear Functions and Forms

Forms of Linear Equations

• Slope-intercept form: $y = mx + b$

• Point-slope form: $y - y_1 = m(x - x_1)$

• Intercept form: $\frac{x}{a} + \frac{y}{b} = 1$

Vertical and Horizontal Lines

• Vertical line: $x = a$

• Horizontal line: $y = b$

Piecewise Linear Function Example

• $g(x) = \begin{cases} x, & x < 2 \\ 5x + 6, & x \geq 2 \end{cases}$

Section Seven: Special Functions and Their Graphs

Absolute Value Function

• $f(x) = |x|$

• Graph is V-shaped, always non-negative.

Even and Odd Power Functions

• Even power: $f(x) = x^n$ for even $n$ (symmetric about y-axis).

• Odd power: $f(x) = x^n$ for odd $n$ (symmetric about origin).

Even and Odd Root Functions

• Even root: $f(x) = \sqrt[n]{x}$ for even $n$ (domain $x \geq 0$).

• Odd root: $f(x) = \sqrt[n]{x}$ for odd $n$ (domain $x \in \mathbb{R}$).

Section Eight: Difference Quotient

Definition

The difference quotient is used to compute the average rate of change of a function and is foundational for derivatives.

• Formula: $\frac{f(x+h) - f(x)}{h}$

• Example: For $f(x) = 3x$, $\frac{f(x+h) - f(x)}{h} = \frac{3(x+h) - 3x}{h} = 3$

Section Nine: Compound Inequalities

Types of Solutions

• Disjoint solution: Two intervals that do not overlap, e.g., $(-\infty, -3) \cup (2, \infty)$

• Intersecting solution: Overlapping intervals, e.g., $[3, 2]$

Interval Notation

• [ ] means closed interval (includes endpoints).

• ( ) means open interval (excludes endpoints).

Section Ten: Function Operations

Sum, Difference, Product, and Quotient

• Given $f(x)$ and $g(x)$, operations are defined as:

  • Sum: $(f+g)(x) = f(x) + g(x)$

  • Difference: $(f-g)(x) = f(x) - g(x)$

  • Product: $(fg)(x) = f(x)g(x)$

  • Quotient: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

Section Eleven: Summary Table of Function Types

Function Type	Example	Graph Shape
Linear	$y = mx + b$	Straight line
Absolute Value	$y = |x|$	V-shape
Quadratic (Even Power)	$y = x^2$	Parabola
Cubic (Odd Power)	$y = x^3$	S-shaped curve
Square Root (Even Root)	$y = \sqrt{x}$	Half parabola
Cube Root (Odd Root)	$y = \sqrt[3]{x}$	S-shaped curve

Additional info:

• These notes cover foundational concepts in calculus, including function definitions, types, domains, ranges, and basic algebraic manipulations. Understanding these is essential for success in calculus and higher mathematics.