General Mathematics: Functions, Graphs, and Piecewise Functions

Foundations of Relations and Functions

• Relation Definition:

  • A relation is a set of ordered pairs that expresses a mathematical relationship or connection between two quantities.
  • The domain represents the complete set of all possible input values ($x$-values).
  • The range represents the complete set of all possible output values ($y$-values).

• Function Definition:

  • A function is a specialized type of relation in which each element of the domain corresponds to exactly one element of the range.
  • It is formally represented in functional notation as $f(x) = y$.
  • Permissible function mappings include:
  • One-to-one mapping (each unique input maps to a unique output).
  • Many-to-one mapping (two or more distinct inputs map to the same output).
  • Impermissible function mappings include:
  • One-to-many mapping (a single input mapping to multiple distinct outputs violates the definition of a function).

• Function Rule:

  • A function rule defines the precise mathematical operation or sequence of operations performed on the input variable $x$ to generate the corresponding output $f(x)$.
  • Example: Given the rule $f(x) = 3x$, the mathematical operation entails multiplying the input value $x$ by $3$.

• Vertical Line Test:

  • A graphical criterion used to identify whether a curve represents a function.
  • If any vertical line intersects a graph at more than one point, the relation is not a function.
  • If every vertical line intersects the graph at most once across its entire domain, the relation is a function.

Key Graph Characteristics and Analysis Techniques

• Domain:

  • Conceptual meaning: The collection of all possible $x$-coordinates for which the function is defined.
  • Determination technique: Scan the horizontal span of the graph from left to right, or solve for the algebraic domain restrictions dictated by the equation (such as preventing division by zero or negative values under even roots).

• Range:

  • Conceptual meaning: The collection of all possible $y$-coordinates resulting from the function's evaluation.
  • Determination technique: Scan the vertical span of the graph from bottom to top, identifying the absolute lowest and highest boundaries.

• $x$-Intercept:

  • Conceptual meaning: The point or points where the graph crosses or touches the horizontal $x$-axis.
  • Determination technique: Set $y = 0$ (or $f(x) = 0$) and algebraically solve for $x$. Expressed as the ordered pair $(x, 0)$.

• $y$-Intercept:

  • Conceptual meaning: The point where the graph crosses or touches the vertical $y$-axis.
  • Determination technique: Set $x = 0$ and evaluate the function $f(0)$. Expressed as the ordered pair $(0, y)$.

• Extrema (Minimum and Maximum):

  • Conceptual meaning: The lowest attained value (minimum) or highest attained value (maximum) of a function.
  • Determination technique: Identify stationary turning points, vertices, or bounded endpoints within the graph's geometry.

Classification and Families of Functions

• Linear Function:

  • Standard algebraic form: $f(x) = mx + b$
  • Geometric graph shape: A non-vertical straight line with constant slope $m$ and vertical shift $b$.

• Absolute Value Function:

  • Standard algebraic form: $f(x) = a|x - h| + k$
  • Geometric graph shape: A V-shaped curve characterized by symmetrical linear arms originating from a distinct vertex point located at $(h, k)$.

• Quadratic Function:

  • Standard algebraic form: $f(x) = a(x - h)^2 + k$
  • Geometric graph shape: A smooth, symmetric U-shaped parabola opening upward (if $a > 0$) or downward (if $a < 0$), featuring a turning vertex at $(h, k)$.

• Square Root Function:

  • Standard algebraic form: $f(x) = a√(x - h) + k$
  • Geometric graph shape: A one-sided curved arc originating at a sharp initial endpoint at $(h, k)$.

• Cube Root Function:

  • Standard algebraic form: $f(x) = a∛(x - h) + k$
  • Geometric graph shape: A continuous S-shaped curve extending infinitely in both directions and pivoting through an inflection center point at $(h, k)$.

Worked Examples: Analysis of Core Function Families

• Linear Function Analysis ($f(x) = 2x + 1$):
  • Domain: $(-\infty, \infty)$, representing all real numbers.
  • Range: $(-\infty, \infty)$, representing all real numbers.
  • $x$-intercept calculation:
  • Set $f(x) = 0$:
  • $0 = 2x + 1$
  • $2x = -1$
  • $x = -\frac{1}{2}$
  • Resulting intercept: $(-0.5, 0)$.
  • $y$-intercept calculation:
  • Set $x = 0$:
  • $f(0) = 2(0) + 1 = 1$
  • Resulting intercept: $(0, 1)$.
  • Extrema:
  • Minimum: None; the line extends infinitely downward.
  • Maximum: None; the line extends infinitely upward.

• Absolute Value Function Analysis ($f(x) = |x - 1| + 1$):
  • Domain: $(-\infty, \infty)$, representing all real numbers.
  • Range: $[1, \infty)$, because the lowest attained vertical value is $y = 1$.
  • $x$-intercept calculation:
  • Set $f(x) = 0$:
  • $|x - 1| + 1 = 0$
  • $|x - 1| = -1$
  • Because an absolute value expression cannot yield a negative output, no real solution exists. Therefore, there is no $x$-intercept.
  • $y$-intercept calculation:
  • Set $x = 0$:
  • $f(0) = |0 - 1| + 1 = |-1| + 1 = 1 + 1 = 2$
  • Resulting intercept: $(0, 2)$.
  • Extrema:
  • Minimum: Attained at the vertex coordinates $(1, 1)$.
  • Maximum: None; both linear arms ascend upward indefinitely.

• Quadratic Function Analysis ($f(x) = -(x - 1)^2 + 1$):
  • Domain: $(-\infty, \infty)$, representing all real numbers.
  • Range: $(-\infty, 1]$, because the parabola opens downward and reaches its maximum boundary at $y = 1$.
  • $x$-intercept calculation:
  • Set $f(x) = 0$:
  • $0 = -(x - 1)^2 + 1$
  • $(x - 1)^2 = 1$
  • $x - 1 = \pm 1$
  • For $x - 1 = 1 \implies x = 2$
  • For $x - 1 = -1 \implies x = 0$
  • Resulting intercepts: $(0, 0)$ and $(2, 0)$.
  • $y$-intercept calculation:
  • Set $x = 0$:
  • $f(0) = -(0 - 1)^2 + 1 = -(1) + 1 = 0$
  • Resulting intercept: $(0, 0)$.
  • Extrema:
  • Minimum: None; the parabola extends downward without bound.
  • Maximum: Attained at the vertex coordinates $(1, 1)$.

• Square Root Function Analysis ($f(x) = \sqrt{x - 1} + 1$):
  • Domain: $[1, \infty)$, because the radicand must remain non-negative:
  • $x - 1 \ge 0 \implies x \ge 1$
  • Range: $[1, \infty)$, because the curve begins at $y = 1$ and increases monotonically.
  • $x$-intercept calculation:
  • Set $f(x) = 0$:
  • $\sqrt{x - 1} + 1 = 0$
  • $\sqrt{x - 1} = -1$
  • Principal square roots are non-negative; therefore, this equation is impossible over the real numbers and there is no $x$-intercept.
  • $y$-intercept calculation:
  • Evaluated at $x = 0$, but $x = 0$ is not contained in the domain $[1, \infty)$. Consequently, there is no $y$-intercept.
  • Extrema:
  • Minimum: Located at the terminal endpoint $(1, 1)$.
  • Maximum: None; the curve continues increasing upward indefinitely.

• Cube Root Function Analysis ($f(x) = \sqrt[3]{x - 1} + 1$):
  • Domain: $(-\infty, \infty)$, representing all real numbers (odd roots accept negative radicands).
  • Range: $(-\infty, \infty)$, representing all real numbers.
  • $x$-intercept calculation:
  • Set $f(x) = 0$:
  • $\sqrt[3]{x - 1} + 1 = 0$
  • $\sqrt[3]{x - 1} = -1$
  • Cube both sides: $x - 1 = (-1)^3 = -1$
  • Solve: $x = 0$
  • Resulting intercept: $(0, 0)$.
  • $y$-intercept calculation:
  • Set $x = 0$:
  • $f(0) = \sqrt[3]{0 - 1} + 1 = \sqrt[3]{-1} + 1 = -1 + 1 = 0$
  • Resulting intercept: $(0, 0)$.
  • Extrema:
  • Minimum: None; the curve descends downward without bound.
  • Maximum: None; the curve ascends upward without bound.
  • Structural center point: Located at the inflection coordinate $(1, 1)$.

Principles and Graphing of Piecewise Functions

• Conceptual Framework:

  • A piecewise function is defined by multiple distinct mathematical sub-functions, each applied across designated sub-intervals of its domain.
  • It derives its name from its graphical presentation, which consists of multiple separate linear or non-linear pieces.

• General Form:

  • Expressed notationally as:     $\begin{cases} f(x) = \text{rule 1}, & \text{interval 1} \\ f(x) = \text{rule 2}, & \text{interval 2} \\ \dots & \dots \end{cases}$

• Step-by-Step Graphing Guidelines:

  • Identify the domain intervals and the exact transition points (breakpoints) where the governing rule alters.
  • Evaluate the sub-function equations at their corresponding breakpoint boundaries to identify plotting anchors.
  • Graph each function rule solely over its restricted interval domain.
  • Use an open circle ($\circ$) at an endpoint when the boundary value is excluded (corresponding to strict inequalities $<$ or $>$).
  • Use a closed circle ($\bullet$) at an endpoint when the boundary value is included (corresponding to non-strict inequalities $\le$ or $\ge$).
  • When two continuous pieces connect at identical boundary coordinates, indicate the point as a solid defined coordinate rather than superimposing open and filled circles.
  • Analyze graph continuity by inspecting whether adjoining pieces connect seamlessly or produce an abrupt jump.

Real-World Applications and Models of Piecewise Functions

• Example 1: Jeepney Fare Model (Continuous):
  • Scenario specifications: The fare costs a flat fee of $14\,\text{PHP}$ for the initial $4\,\text{km}$, after which the fare increases by $2\,\text{PHP}$ for each additional kilometer traveled.
  • Mathematical piecewise formulation:     $F(x) = \begin{cases} 14, & 0 < x \le 4 \\ 14 + 2(x - 4), & x > 4 \end{cases}$
  • Breakpoint analysis: At $x = 4$, the initial rule produces $F(4) = 14$, while the second rule approaches $14 + 2(4 - 4) = 14$.
  • Transition coordinate: $(4, 14)$.
  • Continuity: Because both sub-rules yield an identical value of $14\,\text{PHP}$ at the boundary, the curve connects without separation and is continuous.

• Example 2: Parking Fee Structure (Continuous):
  • Scenario specifications: Parking fees follow a three-tier pricing structure:
  • Flat rate of $50\,\text{PHP}$ for the first $2\,\text{hours}$.
  • Surcharge of $20\,\text{PHP}$ per additional hour for durations between $2\,\text{hours}$ and $5\,\text{hours}$.
  • Surcharge of $15\,\text{PHP}$ per additional hour for parking durations exceeding $5\,\text{hours}$.
  • Mathematical piecewise formulation:     $P(t) = \begin{cases} 50, & 0 < t \le 2 \\ 50 + 20(t - 2), & 2 < t \le 5 \\ 110 + 15(t - 5), & t > 5 \end{cases}$
  • Breakpoint analysis:
  • Transition at $t = 2$: Connecting point is $(2, 50)$.
  • Transition at $t = 5$: Evaluates to $50 + 20(5 - 2) = 50 + 60 = 110\,\text{PHP}$, connecting at $(5, 110)$.
  • Continuity: All adjacent pieces meet at identical coordinates, producing an unbroken continuous piecewise graph.

• Example 3: Seaside Resort Accommodation (Continuous):
  • Scenario specifications: A resort charges a flat fee of $2000\,\text{PHP}$ for a stay of up to $3\,\text{nights}$. Any additional night beyond $3\,\text{nights}$ incurs an incremental cost of $750\,\text{PHP}$ per night.
  • Mathematical piecewise formulation:     $T(x) = \begin{cases} 2000, & 0 < x \le 3 \\ 2000 + 750(x - 3), & x > 3 \end{cases}$
  • Breakpoint analysis: At $x = 3$, the boundary coordinates evaluate to $(3, 2000)$.
  • Continuity: Both functional intervals evaluate to $2000\,\text{PHP}$ at $x = 3$, establishing a continuous graph.

• Example 4: Courier Delivery Fee (Discontinuous):

  • Scenario specifications: Fixed step rates governed by parcel weight categories:
  • Parcels weighing up to $1\,\text{kg}$ ($0 < w \le 1$): Flat fee of $80\,\text{PHP}$.
  • Parcels weighing between $1\,\text{kg}$ and $3\,\text{kg}$ ($1 < w \le 3$): Flat fee of $120\,\text{PHP}$.
  • Parcels weighing between $3\,\text{kg}$ and $5\,\text{kg}$ ($3 < w \le 5$): Flat fee of $180\,\text{PHP}$.
  • Mathematical piecewise formulation:     $D(w) = \begin{cases} 80, & 0 < w \le 1 \\ 120, & 1 < w \le 3 \\ 180, & 3 < w \le 5 \end{cases}$
  • Breakpoint analysis: Transition thresholds occur at $w = 1$ and $w = 3$.
  • Discontinuity mechanism: The fee jumps abruptly from $80\,\text{PHP}$ to $120\,\text{PHP}$ at $w = 1$, and from $120\,\text{PHP}$ to $180\,\text{PHP}$ at $w = 3$. The intervals do not meet, forming a discontinuous step graph.

• Example 5: Printing Promotion Pricing (Discontinuous):

  • Scenario specifications: A printing service charges a baseline rate of $3\,\text{PHP}$ per page for printing up to $20\,\text{pages}$. For volume orders exceeding $20\,\text{pages}$, a promotional pricing scheme applies: a base price of $50\,\text{PHP}$ plus $2\,\text{PHP}$ for each page exceeding $20$.
  • Mathematical piecewise formulation:     $P(n) = \begin{cases} 3n, & 0 < n \le 20 \\ 50 + 2(n - 20), & n > 20 \end{cases}$
  • Breakpoint analysis at $n = 20$:
  • Initial sub-rule evaluation: $P(20) = 3(20) = 60\,\text{PHP}$. Plotted as a solid closed point at $(20, 60)$.
  • Second sub-rule evaluation: As $n \to 20^+$, the function approaches $50 + 2(20 - 20) = 50\,\text{PHP}$. Because $n = 20$ is excluded from this interval, it is plotted as an open circle at $(20, 50)$.
  • Discontinuity mechanism: An abrupt downward vertical jump occurs at $n = 20$, changing from $60\,\text{PHP}$ to $50\,\text{PHP}$. This break renders the function discontinuous.

• Example 6: Bike Rental Pricing (Continuous):
  • Scenario specifications: A bike rental establishment charges a fixed base fee of $60\,\text{PHP}$ for the first hour ($0 < t \le 1$). Beyond the first hour ($t > 1$), the rate increases by an additional $20\,\text{PHP}$ per hour.
  • Mathematical piecewise formulation:     $B(t) = \begin{cases} 60, & 0 < t \le 1 \\ 60 + 20(t - 1), & t > 1 \end{cases}$
  • Breakpoint analysis at $t = 1$:
  • The first rule terminates at $B(1) = 60\,\text{PHP}$.
  • The second rule evaluates to $60 + 20(1 - 1) = 60\,\text{PHP}$.
  • Both pieces connect seamlessly at the coordinate $(1, 60)$.
  • Calculation of representative operational points:
  • $B(1) = 60\,\text{PHP}$
  • $B(3) = 60 + 20(3 - 1) = 60 + 20(2) = 60 + 40 = 100\,\text{PHP}$
  • $B(5) = 60 + 20(5 - 1) = 60 + 20(4) = 60 + 80 = 140\,\text{PHP}$
  • Continuity: Because the two sub-functions intersect at the exact breakpoint coordinate, the graph is continuous.

Analytical Summary

• Standard Algebraic Functions:

  • Begin analysis by classifying the function family (linear, absolute value, quadratic, square root, or cube root).
  • Determine the domain and range by identifying mathematical restrictions (e.g., radicands $\ge 0$) and reviewing vertical/horizontal bounds.
  • Compute intercepts algebraically by solving $f(x) = 0$ for $x$-intercepts and evaluating $f(0)$ for the $y$-intercept.
  • Locate extrema by evaluating curve geometry, vertex locations, or terminal endpoints.

• Piecewise Functions:

  • Partition the domain explicitly according to the defined interval inequalities.
  • Calculate functional outputs at the boundary breakpoints to place closed (included) or open (excluded) coordinate circles.
  • Check whether adjoining functional pieces converge at identical boundary values to determine continuity versus jump discontinuity.