Unit 1: Relations and Functions

Unit 1 formalizes the mathematical language used to describe relationships between sets through the rigorous study of relations and functions. It expands on foundational algebra to introduce specialized real-valued functions such as power, absolute value, signum, and greatest integer functions—exploring their unique properties and graphical behaviors. This unit is essential for mastering functional compositions, inverses, and the application of mathematical modeling to real-world economic and physical systems.

1.1 Relations

• A relation is defined as any set of ordered pairs.
• A relation from set A to set B is any subset of the Cartesian product $A \times B$, which contains pairs $(x, y)$ where $x \in A$ and $y \in B$.
• Ordered pair: A pair $(x, y)$ where the order of elements is significant; $(x, y) = (y, x)$ only if $x = y$.
• Domain (Dom(R)): The set consisting of all the first components (or coordinates) of the elements in a relation.
• Range (Ran(R)): The set consisting of all the second components (or coordinates) of the elements in a relation.
• Relations can be represented through various methods: sets of ordered pairs, diagrams (mapping), graphs on a coordinate plane, algebraic equations, or inequalities.

1.2 Inverse of Relations and Their Graphs

• The inverse of a relation R, denoted $R^{-1}$, is the set of ordered pairs obtained by interchanging the first and second components: $R^{-1} = \{(y, x) : (x, y) \in R\}$.
• The domain of $R^{-1}$ is equal to the range of R.
• The range of $R^{-1}$ is equal to the domain of R.
• The graph of an inverse relation is obtained by reflecting the graph of the original relation across the line $y = x$.

Process: Finding the Inverse of an Equation Algebraically

1. Interchange the variables $x$ and $y$ in the defining equation (e.g., if $y = 3x+6$, write $x = 3y+6$).
2. Solve the new equation for $y$ in terms of $x$ (e.g., $y = \frac{1}{3}x – 2$).
3. State the resulting expression as the inverse relation $R^{-1}$.

1.3 Types of Functions

• A function is a specialized relation in which each object from the domain is related to exactly one object in the range.
• Notation: A function $f$ from set A to set B is written $f: A \rightarrow B$.
• Functional value: If $(x, y) \in f$, it is written as $y = f(x)$, read as “$f$ of $x$“.
• Vertical Line Test: A set of points in a coordinate plane represents a function if and only if no vertical line intersects the graph more than once.

1.3.1 Power Functions

• A power function takes the form $f(x) = ax^r$, where $a \neq 0$ and $r$ are real numbers.
• Properties for natural number $n$:
  • Even $n$: The domain is all real numbers ($\mathbb{R}$), the range is $[0, \infty)$, and the graph is a symmetric parabola-like curve.
  • Odd $n$: Both the domain and range are $\mathbb{R}$, and the graph has a characteristic “S” shape passing through the origin.
• Properties for $f(x) = x^{1/n}$:
  • Even $n$: Domain and range are restricted to $[0, \infty)$.
  • Odd $n$: Both the domain and range are $\mathbb{R}$.

1.3.2 Modulus (Absolute Value) Function

• The absolute value (or modulus) of a number $a$, denoted $|a|$, is its non-negative distance from zero.
• The modulus function is defined as $f(x) = |x|$.
• Properties:
  • $|ab| = |a||b|$ and $|\frac{a}{b}| [cite_start]= \frac{|a|}{|b|}$.
  • Triangular Inequality: $|a + b| \le |a| + [cite_start]|b|$.
• Domain: $\mathbb{R}$; Range: $[0, \infty)$.

1.3.3 Signum Function

• The signum function (sgn x) is a piecewise function that indicates the sign of $x$:
  • $1$ if $x > 0$
  • $0$ if $x = 0$
  • $-1$ if $x < 0$
• Domain: $\mathbb{R}$; Range: $\{-1, 0, 1\}$.

1.3.4 Greatest Integer (Floor) Function

• The greatest integer function (or floor function) maps a real number $x$ to the largest integer less than or equal to $x$, denoted $\lfloor x \rfloor$.
• Domain: $\mathbb{R}$; Range: The set of all integers ($\mathbb{Z}$).
• Standard rule: $\lfloor x \rfloor = m$ (where $m$ is an integer) if and only if $m \le x < m+1$.

Process: Solving Greatest Integer Equations

1. Rewrite the equation $\lfloor f(x) \rfloor = m$ as the inequality $m \le f(x) < m+1$.
2. Use a “change of variable” (e.g., let $z = \lfloor x \rfloor$) for nested floor functions to simplify the solving process.
3. Solve the resulting linear or compound inequalities to find the solution set, typically expressed as an interval.

1.4 Composition of Functions

• Composition occurs when one function is substituted into another: $(g \circ f)(x) = g(f(x))$.
• The domain of $g \circ f$ consists of all $x$ in the domain of $f$ such that the functional value $f(x)$ is within the domain of $g$.

1.5 Inverse Functions and Their Graphs

• A function $f$ is invertible if and only if its inverse is also a function.
• The identity function ($I$) on set A is defined as $f(x) = x$ for all $x \in A$.
• Fundamental property: For an invertible function, $f \circ f^{-1} = I = f^{-1} \circ f$.
• If the composition $(f \circ g)(x) = x$ and $(g \circ f)(x) = x$, then $f$ and $g$ are inverses of each other.

1.6 Applications

• Economic Equilibrium: Calculated by setting total income ($Y$) equal to total expenditure ($E$), where $E$ is the sum of consumption ($C$), investment ($I$), and government spending ($G$).
• Business Modeling: Functions are used to determine revenue, production cost $C(x)$, and profit.
• Variation: Modeling physical laws, such as force on a spring varying directly with the distance stretched.

Key Terminology

• Absolute value: The non-negative magnitude of a real number regardless of its sign.
• Broken line: A dashed line used in graphs to indicate that boundary points are excluded from the solution set.
• Cartesian product: The set of all possible ordered pairs where the first element is from one set and the second from another.
• Composition of functions: The operation of applying one function to the results of another.
• Domain: The set of all input values for which a relation or function is defined.
• First component: The $x$-coordinate in an ordered pair $(x, y)$.
• Function: A relation where every input is associated with exactly one output.
• Greatest integer (floor) function: A function that rounds a number down to the nearest integer.
• Identity function: A function that returns its input unchanged.
• Inverse function: A function that reverses the effect of the original function.
• Invertible: A property of a function that allows for a unique inverse function to exist.
• Modulus: A synonym for the absolute value.
• Ordered pairs: A set of two elements where the order $(x, y)$ signifies a specific coordinate.
• Power function: A function defined by a variable base raised to a fixed real power.
• Range: The set of all possible output values of a relation or function.
• Relation: A general set of ordered pairs connecting two sets.
• Second component: The $y$-coordinate in an ordered pair $(x, y)$.
• Signum (sgn) function: A piecewise function that outputs the sign $(-1, 0, or 1)$ of a number.
• Solid line: A continuous line used in graphs to indicate that boundary points are included in the solution set.
• Triangular Inequality: The mathematical property stating the absolute value of a sum is less than or equal to the sum of absolute values.
• Vertical line test: A visual check to confirm if a graph represents a function.