Mathematical Functions and Notation for Scientists: Foundations and Applications

Mathematical Notation

Sets of Numbers

Mathematics for scientists often begins with a review of the fundamental sets of numbers used in analysis and modeling.

• Natural Numbers (ℕ): The set of positive integers used for counting. ℕ = {1, 2, 3, 4, ...}

• Integers (ℤ): The set of whole numbers, including negatives, zero, and positives. ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}

• Real Numbers (ℝ): The set of all numbers on the number line, including rationals and irrationals.

Intervals

Intervals are used to describe subsets of real numbers:

• Closed Interval [a, b]: All x such that $a \leq x \leq b$

• Open Interval (a, b): All x such that $a < x < b$

• Half-Open Intervals: [a, b): $a \leq x < b$ (a, b]: $a < x \leq b$

• Intervals may extend to infinity, e.g., (0, ∞) for non-negative real numbers.

Functions

Definition and Properties

A function is a rule that assigns each input exactly one output. Formally, a function f from set A to set B is a mapping such that every element of A is assigned to a unique element of B.

• Input: Independent variable

• Output: Dependent variable

• Each input has exactly one output (no input maps to multiple outputs)

Example: The kinetic energy of a mass m moving at velocity v is a function: $E(v) = \frac{1}{2}mv^2$

Representations of Functions

Functions can be represented in several ways:

• Table: Lists input-output pairs.

  x	1	1.5	2	2.5	3	3.5	4
  y	0	1.25	3	5.25	8	11.25	15

• Algebraic Formula: Expresses the rule using symbols. e.g., $y = x^2 - 1$

• Graph: Plots the function on a coordinate system.

• Verbal Description: Describes the rule in words.

Changing Symbols

The symbols used for input and output can be changed without altering the function's rule.

• $f(x) = \frac{x-1}{2}$

• $g(t) = \frac{t-1}{2}$

• Both define the same function.

Evaluating Functions

To find the output for a specific input, substitute the value into the function's formula.

• Example: $f(x) = \frac{2x^2 - 4x + 3}{x - 2}$ Calculate $f(-2)$, $f(0)$, $f(3)$ by substituting each value for x.

Functions in Science

Functions are used to model relationships between physical quantities.

• Kinetic Energy: $E(v) = \frac{1}{2}mv^2$

• Spring Force: $F(x) = -kx$

Vertical Line Test

A curve in the xy-plane is the graph of a function if and only if no vertical line crosses it more than once.

• Example: The circle $x^2 + y^2 = 9$ fails the vertical line test, so it is not a function.

• Algebraically, $y^2 = 9 - x^2$ gives $y = \pm\sqrt{9 - x^2}$, which is not unique for each x.

Special Functions

Square Root Function

The square root of x is a number y such that $y^2 = x$.

• For $x > 0$, there are two square roots: $\sqrt{x}$ and $-\sqrt{x}$.

• For $x = 0$, $\sqrt{0} = 0$.

• For $x < 0$, there is no real square root.

• The square root function is defined as $f(x) = \sqrt{x}$ for $x \geq 0$.

• Only returns non-negative values.

Absolute Value Function

The absolute value of x, denoted $|x|$, is defined as:

• $|x| = x$ if $x \geq 0$

• $|x| = -x$ if $x < 0$

• This is a piecewise-defined function.

Piecewise-Defined Functions

Definition and Example

Piecewise functions are defined by different expressions over different intervals.

• Example: $f(x) = \begin{cases} x, & -2 < x < 2 \\ -2, & x > 2 \end{cases}$ The function is continuous at $x = 2$ but makes a jump, so it is discontinuous at $x = 2$.

Domain and Range of Functions

Definitions

• Domain: The set of all input values (x) for which the function is defined.

• Range: The set of all possible output values $f(x)$ for x in the domain.

Domains may be specified explicitly or determined by the formula's restrictions.

• Example: $f(x) = x^2$ on $[1,2]$ Domain: $[1,2]$ Range: $[1,4]$

• Example: $f(x) = x^2$ on $[-1,1]$ Domain: $[-1,1]$ Range: $[0,1]$

• Example: $f(x) = \sqrt{x}$ Domain: $[0, \infty)$ Range: $[0, \infty)$

Algorithm for Finding Domain

• For $\sqrt{g(x)}$, require $g(x) \geq 0$.

• For $\frac{1}{g(x)}$, require $g(x) \neq 0$.

• For trigonometric functions like $\tan(x)$ and $\cot(x)$, exclude points where the denominator is zero.

• Domain is the intersection of all restrictions.

Examples of Domain and Range

• Example: $f(x) = 2 + \sqrt{x - 1}$ Domain: $[1, \infty)$ Range: $[2, \infty)$

• Example: $f(x) = \frac{x+1}{x-1}$ Domain: $\mathbb{R} \setminus \{1\}$ Range: $(-\infty, 1) \cup (1, \infty)$

Combining Functions: Arithmetic Operations

Sum, Difference, Product, and Quotient

Functions can be combined using arithmetic operations:

• $(f+g)(x) = f(x) + g(x)$

• $(f-g)(x) = f(x) - g(x)$

• $(fg)(x) = f(x)g(x)$

• $(f/g)(x) = \frac{f(x)}{g(x)}$ (where $g(x) \neq 0$)

• The domain of the combined function is the intersection of the domains of $f$ and $g$ (and any additional restrictions).

Examples of Combining Functions

• Example: $f(x) = x+2$, $g(x) = x+1$ $(f+g)(x) = 2x+3$

• Example: $f(x) = \sin x$, $g(x) = x-1$ Domain: $x > 2$ and $x < 5$

• Example: $f(x) = 1 + \sqrt{x-2}$, $g(x) = x-3$ $(f+g)(x) = x-2 + \sqrt{x-3}$ Domain: $[3, \infty)$

• Example: $f(x) = x$, $g(x) = 1/x$ $(f/g)(x) = x^2$ Domain: $(-\infty, 0) \cup (0, \infty)$

Function Composition

Definition

The composition of two functions $f$ and $g$ is denoted $f \circ g$ and defined by $(f \circ g)(x) = f(g(x))$.

• The domain of $f \circ g$ is the set of all x in the domain of $g$ such that $g(x)$ is in the domain of $f$.

Examples of Composition

• Example: $f(x) = x$, $g(x) = 2x+1$ $(f \circ g)(x) = g(x) = 2x+1$

• Example: $f(x) = x^2 + 3$, $g(x) = \sqrt{x}$ $(f \circ g)(x) = (\sqrt{x})^2 + 3 = x + 3$ Domain: $[0, \infty)$

• Example: $(g \circ f)(x) = \sqrt{x^2 + 3}$ Domain: $(-\infty, \infty)$ Range: $[\sqrt{3}, \infty)$

Set Operations

Intersection and Union

Domains and ranges can be described using set operations:

• Intersection ($A \cap B$): Points in both sets.

• Union ($A \cup B$): Points in either set.

Summary Table: Function Types and Properties

Function Type	Definition	Domain	Range
Square Root	$f(x) = \sqrt{x}$	$[0, \infty)$	$[0, \infty)$
Absolute Value	$f(x) = |x|$	$(-\infty, \infty)$	$[0, \infty)$
Quadratic	$f(x) = x^2$	$(-\infty, \infty)$	$[0, \infty)$
Reciprocal	$f(x) = 1/x$	$(-\infty, 0) \cup (0, \infty)$	$(-\infty, 0) \cup (0, \infty)$
Piecewise	Defined by cases	Depends on definition	Depends on definition

Additional info: These foundational concepts in functions and their properties are essential for modeling and analyzing physical systems in physics and other sciences. Understanding domains, ranges, and function operations is critical for interpreting scientific formulas and graphs.