Calculus 1: Relations and Functions Study Notes on Functions and Relations

Origins and Foundations of Calculus

Calculus is a branch of mathematics concerned with the study of rates of change. It was developed during the latter half of the $17^{th}$ century by two primary figures: Gottfried Leibniz and Isaac Newton. Isaac Newton was the first to develop calculus, applying the principles directly to the understanding of physical systems. Independently, Gottfried Leibniz developed the notations that are widely used in the field today. While basic mathematics relies on operations such as addition ($+$), subtraction ($-$), multiplication ($\times$), and division (\÷), calculus employs operations involving functions and integrals to determine rates of change in real-time.

Historically, mathematics was static before the invention of calculus, capable only of calculating objects that were perfectly still. However, the universe is in a constant state of motion, from subatomic particles and biological cells to stars and galaxies in space. Calculus provides the tools necessary to determine how particles and matter actually move and change, making it essential for understanding a universe that is never truly at rest.

Branches of Calculus: Differential and Integral

There are two primary branches of calculus: differential calculus and integral calculus. Differential calculus focuses on the rate of change of functions with respect to their variables, predominantly through the use of derivatives and differentials. In graphical terms, a derivative represents the slope of a line, which is calculated as the rise over the run. This branch focuses on the curve itself and its immediate properties.

In contrast, integral calculus seeks to find the total quantity when the rate of change is already known. It focuses on concepts such as the slopes of tangent lines and velocities, but its primary concern is the space or area under a curve. Integral calculus is employed to calculate total size or values, including lengths, areas, and volumes. These two branches allow marine engineering and naval architecture students to analyze both the curve of a ship's hull (differential calculus) and the volume or area under that hull (integral calculus).

Practical Applications of Calculus

Calculus has a vast array of practical applications across numerous scientific and engineering disciplines. In physics, it is used to define and calculate motion, electricity, heat, light, harmonics, acoustics, astronomy, and dynamics. Albert Einstein's theory of relativity relies heavily on calculus. In chemistry, it is used to calculate the rates of radioactive decay, while in biology and sociology, it helps predict birth and death rates.

Calculus is also integral to geography, computer vision (essential for autonomous vehicle driving), photography, artificial intelligence, robotics, video games, and film production. In engineering, it is utilized for bridge design, analyzing fluid flow, studying planetary motion and gravity, and geometric curve design. Within the specific field of shipbuilding, calculus has been used for years to determine hull curves and general ship design. In economics, it assists in predicting the potential profits of companies or entire industries.

The Rule of Correspondence and Relations

A rule of correspondence is any equation that describes how the elements of a domain and range in a relation are paired; it effectively determines the range. A relation is defined as a set of ordered pairs. The nature of a relation depends on its domain and the rule applied to it.

Consider Relation $A = \{(x, y) | y = x + 2, x = 2, 4, 6\}$. This is a set of specific ordered pairs: $(2, 4)$, $(4, 6)$, and $(6, 8)$. The domain is the set $\{2, 4, 6\}$ and the range is the set $\{4, 6, 8\}$. The graph of this relation consists of only three discrete points.

Relation $B = \{(x, y) | y = x + 2, x, y \in \mathbb{R}\}$ consists of an infinite number of ordered pairs because the domain includes all real numbers. When a domain is not explicitly indicated, it is assumed to consist of all real numbers. The graph of relation $B$ is a line extending indefinitely. The domain is $\{x | x \in \mathbb{R}\}$ and the range is $\{y | y \in \mathbb{R}\}$.

Relation $C = \{(x, y) | y = x + 2, x \neq -2\}$ indicates that the domain consists of all real values except $x = -2$. Consequently, the point $(-2, 0)$ is excluded from the set. On a graph, this is represented by an open circle around that point. The domain is $\{x | x \neq -2\}$ and the range is $\{y | y \neq 0\}$.

Mathematical Definition of a Function

A function is a specialized type of relation consisting of ordered pairs $(x, y)$ such that no two pairs have the same first element $x$. Every element in the domain must be associated with exactly one element in the range. Relations $A$, $B$, and $C$ mentioned previously are all functions because every value of $x$ corresponds to one and only one value of $y$.

The Vertical Line Test is a method used to determine if a relation is a function based on its graph. If every vertical line drawn through the graph intersects it at exactly one point, the relation is a function. If any vertical line crosses the graph at more than one point, the relation is not a function (i.e., $y$ is not a function of $x$).

Constants, Variables, and Function Notation

In mathematics, a constant is a quantity that maintains a fixed value within a problem. Absolute constants, such as $\pi ≈ 3.14159$, $e ≈ 2.71828$, $\sqrt{2}$, and integers like $3$, retain their values across all mathematical contexts. Arbitrary constants remain fixed within one specific problem but can take different values in other problems.

A variable is a quantity that can assume various values during a problem. In the equation $y = 1 - x$, $x$ is the independent variable because its values are freely assumed, and $y$ is the dependent variable because its value relies on the value of $x$.

Function notation uses letters such as $f$, $g$, $H$, $M$, or Greek letters like $\alpha$ and $\beta$ to represent specific relationships. The notation $f(x)$ is read as "function of $x$." For example, in the equation $y = -\sqrt{x+1}$, we can write $y = \alpha(x)$, meaning $y$ is a function of $x$. In this case House, the ordered pair is $[x, \alpha(x)]$ or $(x, -\sqrt{x+1})$. This is a single-valued function. However, the notation $f(x) = \sqrt{x+1}$ can be a double-valued function; if $x = 3$, then $f(3) = \pm 2$. Functions can also depend on multiple independent variables, such as $\beta(x, z)$, which is read as "function $\beta$ of $x$ and $z$."

Function Evaluation and Algebraic Procedures

Function evaluation is the process of finding the specific value of a function given a value for the independent variable. For instance, $h(-1)$ represents the value of function $h$ when $x = -1$. In a multivariable function $f(x, y)$, $f(-2, 4)$ signifies the value of the function when $x = -2$ and $y = 4$.

Example 4: Define $f(x) = x^2 - 2x - 3$. Evaluating at various points:

• $f(0) = (0)^2 - 2(0) - 3 = -3$
• $f(-1) = (-1)^2 - 2(-1) - 3 = 1 + 2 - 3 = 0$
• $f(-2) = (-2)^2 - 2(-2) - 3 = 4 + 4 - 3 = 5$
• $f(1) = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4$
• $f(2) = (2)^2 - 2(2) - 3 = 4 - 4 - 3 = -3$
• $f(3) = (3)^2 - 2(3) - 3 = 9 - 6 - 3 = 0$
• Piecewise composition Evaluation: $f[f(x)] = f(x^2 - 2x - 3) = (x^2 - 2x - 3)^2 - 2(x^2 - 2x - 3) - 3$
• Expanding the substitution results in: $x^4 + 4x^2 + 9 - 4x^3 - 6x^2 + 12x - 2x^2 + 4x + 6 - 3 = x^4 - 4x^3 - 4x^2 + 16x + 12$.

Example 5: Calculating the difference quotient for $f(x) = \frac{1}{x^2}$. First evaluate $f(x+h) = \frac{1}{(x+h)^2}$. Then subtract $f(x)$: $\frac{f(x+h) - f(x)}{h} = \frac{\frac{1}{(x+h)^2} - \frac{1}{x^2}}{h} = \frac{\frac{x^2 - (x+h)^2}{x^2(x+h)^2}}{h} = \frac{x^2 - (x^2 + 2hx + h^2)}{hx^2(x+h)^2} = \frac{-2hx - h^2}{hx^2(x+h)^2} = \frac{-h(2x+h)}{hx^2(x+h)^2} = \frac{-(h+2x)}{x^2(x+h)^2}$.

Domain and Range with Implicit Restrictions

If the domain of a function is not explicitly stated, one must identify it based on mathematical constraints, such as the fact that division by zero is undefined and the square root of a negative number is not a real number.

In Example 6, the functions $H(x) = \frac{9-x^2}{3+x}$ and $G(x) = 3-x$ appear identical because $9-x^2$ factors into $(3-x)(3+x)$. However, $G(x)$ is defined for all real numbers ($x \in \mathbb{R}$), while $H(x)$ is undefined at $x = -3$ because the denominator becomes zero. Graphically, $H(x)$ has an open circle at the point $(-3, 6)$.

In Example 7, for $B(x) = \sqrt{x-4}$, the radicand must be non-negative: $x-4 \geq 0 \rightarrow x \geq 4$. The domain is $\{x | x \geq 4\}$. Since the principal square root is always zero or positive, the range is $\{y | y \geq 0\}$.

In Example 8, for $f(x) = x^2 + 2$ on the restricted interval $-2 < x ≤ 1$, the domain is specified as $\{x | -2 < x \leq 1\}$. Calculating boundary values: when $x = 1$, $f(1) = 3$; as $x \rightarrow -2$, $f(-2) \rightarrow 6$. The vertex of the parabola is at $(0, 2)$. Therefore, the range is $\{y | 2 \leq y < 6\}$.

Further examples of implicit domains include $f(x) = \frac{x^2-4}{x-2}$, which is undefined at $x = 2$. For $h(x) = \sqrt{1-x^2}$, the domain consists of values where $1-x^2 \geq 0$, resulting in the interval $-1 ≤ x ≤ 1$. The graph is the upper half of a circle centered at the origin with a radius of $1$.

Piecewise-Defined Functions

A piecewise-defined function is a function where the domain is divided into distinct parts, with each part defined by a different rule or formula. A property is described as "piecewise" if it holds for each individual piece but not necessarily for the entire domain. The absolute value function is a classic piecewise-defined function: $f(x) = |x| = \begin{cases} -x, & x < 0 \\ 0, & x = 0 \\ x, & x > 0 \end{cases}$

Example 9: Given $f(x) = \begin{cases} x+1, & x > 0 \\ x-2, & x \leq 0 \end{cases}$ The domain is all real numbers $\{x | x \in \mathbb{R}\}$. Evaluation:

• If $x = -3$, $f(-3) = -3 - 2 = -5$.
• If $x = 6$, $f(6) = 6 + 1 = 7$. Graphing shows the range is $\{y | y \leq -2 \cup y > 1\}$.

Example 10: Given $H(x) = \begin{cases} x^2-1, & x > 1 \\ 1, & x = 0 \\ x^2+2, & x < -1 \end{cases}$ The domain is $\{x | x > 1 \cup x = 0 \cup x < -1\}$. Based on the graph, the range is $\{y | y > 3 \cup y = 1 \cup y < 0\}$.

Example 11: For a function $H(x)$ with a domain of all real numbers ($x \in \mathbb{R}$) and a range of $\{y | y \geq 3 \cup y = -2\}$, the specific value of the function at $x = 2$ is identified from its graph as $H(2) = -2$.