Calculus: Functions, Domain, and Range

Thomas' Calculus: Early Transcendentals

• The textbook is titled "Thomas' Calculus: Early Transcendentals"
• The edition is the fifteenth.
• The units are in SI (Système International) units.
• The authors are Hass, Heil, Bogacki, and Weir.
• The book is published by Pearson.

Chapter 1: Functions

• The first chapter of the book is on functions.
• Section 1.1 is titled "Functions and Their Graphs."
• Examples include 2, 4, 7, and 8.
• Exercises include 3, 5, 6 (only domains), 15, 18, and 55.

Functions; Domain and Range

• A function maps elements from set D (domain) to set Y (range).
• A unique value (f(x)) in Y is assigned to each x in D.
• Domain is the set of all possible values of x.
• Range is the set of all possible values of f(x) as x varies throughout the domain.
• Functions can be represented as machines, with inputs (domain values) and outputs (range values).
• A function assigns a single unique value from the range set Y to each element from the domain set D.

Graphs of Functions

• Example 2 involves graphing y = x² over the interval [-2, 2].
• Key points on the graph include (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4).

The Vertical Line Test for a Function

• A curve in the xy-plane represents a function of x if no vertical line intersects the curve more than once.

Piecewise-Defined Functions

• A function can be defined by multiple formulas over different parts of its domain.
• The absolute value function is an example.

Properties of Absolute Values

• |x| = |-x|
• |x| ≤ a implies -a ≤ x ≤ a
• |x| ≥ a implies x ≥ a or x ≤ -a
• |x| = a implies x = ±a or x=a
• √a² = a if a ≥ 0 and √a² = -a if a < 0

Increasing and Decreasing Functions

• A function is increasing on interval I if f(x₁) < f(x₂) when x₁ < x₂ within I.
• A function is decreasing on interval I if f(x₁) > f(x₂) when x₁ < x₂ within I.

Even Functions and Odd Functions

• f(-x) = f(x) is an even function.
• The graph is symmetric about the y-axis.
• Example: f(x) = x²
• f(-x) = -f(x) is an odd function.
• The graph is symmetric about the origin.
• Example: f(x) = x³

Special Properties

• Even + Even = Even
• Odd + Odd = Odd
• Odd + Even = Neither
• Even × Even = Even
• Odd × Odd = Even
• Odd × Even = Odd

Algebraic Functions

• Polynomial functions.
• Rational functions.
• Radical functions.

Common Functions

• Linear Functions: f(x) = mx + b
• Constant Function: f(x) = b
• Identity Function: f(x) = x
• Graphs of lines passing through the origin and constant functions are plotted.

Power Function

• f(x) = x^a where a is constant
• Positive integer values are plotted, with odd values showing symmetry and even values having different graphs in the plots with corresponding domains and ranges.
• Negative integer values of "a" are also plotted, showing reciprocal functions with corresponding domain and range.

Radical Function

• A radical function is an algebraic function f(x) = xⁿ when n is a positive rational index.
• The domain of a radical function depends on whether n is an odd or even number.

Polynomial Function

• A polynomial function is defined by f(x) = a_nxⁿ + ... + a₁x + a₀ where n is a non-negative integer, and the coefficients are real numbers.
• The domain is all real numbers.
• The degree of the polynomial is n.

Rational Functions

• A rational function is defined as the ratio of two polynomial functions.
• The domain excludes values that make the denominator zero.

Homework

• Problems are assigned: 1, 2, 17, 19, 49, 53.

Transcendental Functions

• Functions not algebraic are transcendental functions.
• Trigonometric functions (sine, cosine).
• Exponential functions (e^x).
• Logarithmic functions (log_ax).

Explore functions, domains, and ranges, where a function maps elements from set D (domain) to set Y (range), assigning a unique value f(x) in Y to each x in D. Covers the textbook "Thomas' Calculus: Early Transcendentals".