Functions, Relations, Domain, Range, and Toolkit Functions

Fundamentals of Relations and Functions

• Relations and Coordinate Representation:

  • A mathematical relation is defined as a collection or set of ordered pairs of real numbers, written in the form $(x, y)$.

  • Each ordered pair $(x, y)$ corresponds to a unique geometric point plotted within the two-dimensional Cartesian coordinate system.

  • Plotting isolated points is a basic fallback method for graphing. While useful when no other information is available, connecting discrete points can lead to errors by failing to display fine local curves, oscillations, or discontinuities.

• Tutorial and Class Details:

  • Weekly tutorials consist of five distinct sections running for $50$\,minutes per section.

  • Practice problems from the textbook include Exercise $54$ (worked through directly) and Exercise $53$ (assigned for independent practice).

Evaluating Functions Graphically and the Vertical Line Test

• Evaluating and Solving Functions via Graphs (Exercise 54):

  • An algebraic formula $f(x)$ allows function evaluation by substituting an $x$-value directly into the formula to find $y$, or solving for $x$ by setting the expression equal to a specific $y$-value.

  • Graphical evaluation follows identical principles using visual coordinates:

  • Evaluating $f(4)$:

    • Locate $x = 4$ on the horizontal axis.

    • Inspect vertically (up and down along the line $x = 4$) to locate the point where it intersects the curve.

    • The graph yields a corresponding vertical position of $y = -3$. Thus, $f(4) = -3$.

  • Solving $f(x) = 1$:

    • Locate $y = 1$ on the vertical axis (a horizontal line at height $1$ above the horizontal axis).

    • Identify all points along the graph that intersect this horizontal line.

    • The graph intersects at $x = 0$ and $x = -6$.

    • Thus, the solutions to $f(x) = 1$ are $x = 0$ and $x = -6$, demonstrating that multiple distinct input values $x$ can produce the same output value $y$.

• The Vertical Line Test (VLT):

  • A graph plotted in the Cartesian coordinate system represents $y$ as a function of $x$ if and only if no vertical line intersects the graph more than once.

  • If any vertical line cuts through the graph at two or more points, there exists an input $x$ associated with multiple outputs $y$, violating the definition of a function.

  • Example: $x = y^2$:

  • Includes sample ordered pairs: $(0, 0)$, $(1, 1)$, $(1, -1)$, $(2, \text{sqrt}(2))$, and $(2, -\text{sqrt}(2))$.

  • Plotting these points produces a horizontal parabola opening to the right.

  • Applying vertical lines through positive values of $x$ (such as $x = 1$) results in two intersection points ($y = 1$ and $y = -1$).

  • Conclusion: $x = y^2$ is a valid relation, but $y$ is not a function of $x$.

  • Application to Textbook Exercises 40, 44, and 46:

  • Exercise 40: A curve oscillating up and down that intersects individual vertical lines at two separate points fails the VLT and is not a function.

  • Exercise 44: A continuous curve moving across the plane with end arrows indicating ongoing path behavior intersects every vertical line at most once, passing the VLT and is a function.

  • Exercise 46: A straight, non-vertical line intersects every vertical line exactly once, passing the VLT and is a function.

Coordinate Axis Conventions and the Horizontal Line Test

• The Horizontal Line Test (HLT):

  • While the Vertical Line Test determines whether a relation is a function, the Horizontal Line Test determines whether a function is one-to-one (injective).

  • A function is one-to-one if every output value $y$ corresponds to exactly one input value $x$.

  • If a horizontal line intersects a function's graph more than once, multiple inputs share the same output, meaning the function is not one-to-one.

• Disciplinary Axis Conventions:

  • Standard mathematical, physical, engineering, chemical, and biological convention places input variables ($x$) on the horizontal axis and output variables ($y$) on the vertical axis.

  • As a result, the VLT checks outputs per input (function status), while the HLT checks inputs per output (one-to-one status).

  • Specific fields—namely Economics and Ocean Sciences—invert this convention by placing input variables on the vertical axis and output variables on the horizontal axis.

Interval Notation and Set Representations

• Domain and Range Concepts:

  • Domain: The complete set of all valid input values ($x$-values) for which a function is defined.

  • Range: The complete set of all possible output values ($y$-values) produced by the function.

• Finite Ordered Pair Example:

  • Given the discrete relation defined by the set of pairs: $\{(0.5, 2), (2, 1.5), (3, -0.5), (3.5, 2)\}$.

  • Domain: The set of first coordinates $\{0.5, 2, 3, 3.5\}$.

  • Range: The set of second coordinates $\{-0.5, 1.5, 2\}$ (duplicate outputs are listed only once in set notation).

  • Mapping visualization: Two input elements ($0.5$ and $3.5$) map to the single output element $2$. This relation passes the VLT and is a function, but fails the HLT and is not one-to-one.

• Interval Notation Guidelines:

  • Open Endpoints: Indicated by strict inequalities ($<$ or $>$) and round parentheses $( \text{ or } )$. Endpoints are excluded.

  • Closed Endpoints: Indicated by non-strict inequalities ($\text{le}$ or $\text{ge}$) and square brackets $[ \text{ or } ]$. Endpoints are included.

  • Interval Notation Examples:

  • Inequality $5 < h < 10$ corresponds to the open interval $(5, 10)$.

  • Inequality $5 \text{le} h < 10$ corresponds to the half-open interval $[5, 10)$.

  • Parentheses notation $(a, b)$ used for open intervals shares visual formatting with Cartesian point coordinates $(x, y)$; context dictates the meaning.

Algebraic Determination of Domain

To find the domain of an algebraically defined function, identify and exclude any values of $x$ that cause forbidden operations. The primary algebraic restrictions are:

1. Division by zero (denominator equal to $0$).

2. Even roots (such as square roots) of negative numbers.

• Example 1: Polynomial Function

  • Formula: $f(x) = x^2 - 7$

  • Analysis: No fractions or roots exist. Any real number can be squared and reduced by $7$.

  • Domain: All real numbers, expressed as $\text{mathbb}{R}$ or in interval notation as $(-\text{infty}, \text{infty})$.

• Example 2: Rational Function

  • Formula: $g(x) = \text{frac}{x + 1}{2 - x}$

  • Analysis: The denominator cannot equal zero.

  • Algebraic Condition: $2 - x \text{neq} 0 \text{implies} x \text{neq} 2$

  • Numerator check: Setting $x = -1$ yields $g(-1) = \text{frac}{0}{3} = 0$, which is an allowed real number.

  • Input check for $x = 0$: Yields $g(0) = \text{frac}{1}{2}$, which is an allowed real number.

  • Domain: All real numbers except $x = 2$.

  • Interval Notation: $(-\text{infty}, 2) \text{cup} (2, \text{infty})$, where $\text{cup}$ denotes the set union operator.

• Example 3: Simple Radical Function

  • Formula: $h(x) = \text{sqrt}(2x)$

  • Analysis: The expression inside the square root must be non-negative.

  • Algebraic Condition: $2x \text{ge} 0 \text{implies} x \text{ge} 0$

  • Input check for $x = 0$: Yields $h(0) = \text{sqrt}{0} = 0$, which is allowed.

  • Domain: $[0, \text{infty})$ or $\{x \text{in} \text{mathbb}{R} \text{mid} x \text{ge} 0\}$.

• Example 4: Composite Radical Function

  • Formula: $r(x) = 29 + \text{sqrt}(2 + 3x)$

  • Analysis: The expression under the radical must be non-negative.

  • Algebraic Steps:

  1. Set radicand non-negative: $2 + 3x \text{ge} 0$

  2. Subtract $2$ from both sides: $3x \text{ge} -2$

  3. Divide by $3$: $x \text{ge} -\text{frac}{2}{3}$

  • Domain: $[-\text{frac}{2}{3}, \text{infty})$ or $\{x \text{in} \text{mathbb}{R} \text{mid} x \text{ge} -\text{frac}{2}{3}\}$.

Graphical Determination of Domain and Range

• Visual Projection Technique:

  • Domain: Project ("smash") all points of the graph vertically onto the horizontal $x$-axis. The interval of covered $x$-values forms the domain.

  • Range: Project ("smash") all points of the graph horizontally onto the vertical $y$-axis. The interval of covered $y$-values forms the range.

• Textbook Figure 8 Example:

  • Graph Description: Starts at a solid point at $(-5, 5)$, curves downward through the x-axis, wiggles, and extends indefinitely downward and to the right, terminating in an arrow.

  • Domain Reading: The leftmost $x$-value is $x = -5$ (included) and extends continuously to the right toward $\text{infty}$.

  • Domain Interval: $[-5, \text{infty})$

  • Range Reading: The highest peak reaches $y = 5$ (included) and the arrow extends downwards toward $- ext{infty}$.

  • Range Interval: $(-\text{infty}, 5]$

• Interval Ordering Constraint:

  • Intervals must always be written from the smaller (more negative) number to the larger (more positive) number.

  • Writing a range as $[5, -\text{infty})$ is mathematically incorrect; it must be written as $(-\text{infty}, 5]$.

Toolkit Functions

Toolkit functions represent standard baseline functions in mathematics (referenced in Section 1.1, Table 13 of the textbook).

• Constant Functions:

  • Equation: $f(x) = c$, where $c$ is any real constant (e.g., $c = 2$, $c = 7$, $c = -\text{pi}$).

  • Graph: A flat horizontal line positioned at height y = c$.\n * Domain: (- ext{infty}, ext{infty}).\n * Range: The single-element set {c}.\n * Mapping Property: Extremely non-one-to-one, as every input produces identical output c$.

Student Questions and Clarifications

• Question regarding evaluation of zero in rational function denominators vs. numerators:

  • Clarification: Inputting $x = 0$ into $g(x) = \text{frac}{x+1}{2-x}$ produces $g(0) = \text{frac}{1}{2}$, which is completely valid. A numerator evaluating to $0$ yields an output of $0$ (e.g., $g(-1) = 0$), which is also valid. Only zero in the denominator causes an undefined error.

• Question on formatting domain and range answers on tests:

  • Clarification: If an exam question explicitly demands inequality notation or interval notation, that specific format must be followed. If no format is specified, any mathematically correct representation (set-builder, interval notation, or descriptive words) will receive full credit.

• Question on writing reverse interval bounds (e.g., writing range as $[5, -\text{infty})$):

  • Clarification: Intervals must strictly progress from lower bounds to upper bounds. Reversing the order violates notation standard and results in point deductions.