Functions: definitions, representations, domains & types

Definition of a Function

• Formal statement
  • “A function is a rule that assigns to every element of a set $D$ (domain) exactly one element of a set $E$ (codomain).”
• Machine/Rule metaphor
  • Pick an input $x\in D$ → apply the rule → obtain the unique output $f(x)\in E$.
• Immediate consequences
  • No element of $D$ is left without an image.
  • No element of $D$ can map to two (or more) distinct outputs.
  • It is permissible for two (or more) different inputs to share the same output (e.g.
    $(f(1)=f(-1)=1)$ for $f(x)=x^2$).

Representations of Functions

• Four common ways were discussed. Each representation must honor the two fundamental rules above.

1. Arrow (Mapping) Diagrams

• Visual pairing of elements in two discrete sets.
• Checks:
  • Every element in domain has an outgoing arrow.
  • No domain element has two outgoing arrows.
  • Multiple arrows may converge on the same codomain element.
• Unused codomain elements are acceptable.

2. Tables / Charts

• Two-column list (input | output) — e.g. “Name → Student ID”.
• Good for finite datasets.
• Limitations
  • Impractical for infinite domains (e.g. all real numbers).
  • Still must ensure uniqueness of output per input.

3. Graphs in the $xy$-plane

• Graph = set of ordered pairs $(x, f(x))$.
• Vertical tracing: for a specific $x$, move vertically to the curve to read $f(x)$.
• Deciding if a curve is a function → Vertical Line Test (VLT)
  • “Every vertical line intersects the graph at most one point.”
  • Failure (≥2 intersections) ⇒ some $x$ value would have ≥2 outputs ⇒ not a function.
• Graphs are more general than tables/diagrams because they can implicitly encode infinitely many points.

4. Formulas (Algebraic Rules)

• Most compact & universal description.
• Any expression of the form $f(x)=\text{(some expression involving only }x\text{)}$ qualifies.
• Examples
  • $(f(x)=x^2),\; (f(x)=x+5),\; (f(x)=2),\; (f(x)=(x+5)^2)$.
• Non-example
  • $x\cdot y=1$ (not solved for a unique $y$ in terms of $x$).

Domain, Codomain & Example of Domain Determination

• Terminology
  • Domain (set $D$): all inputs for which the formula “makes sense”.
  • Codomain (set $E$): the target set we declare; may include unrealised values.
• Worked example
  • $f(x)=x^2+\dfrac{1}{\sqrt{x^2-1}}$
  • Constraints
  1. Square–root argument non-negative: $x^2-1\ge0$.
  2. Denominator non-zero: $x^2-1\ne0$.
  • Combine ⇒ $x^2-1>0 \;\Rightarrow\; |x|>1.$
  • Solution intervals:
    $(-\infty,-1)\;\cup\;(1,\infty).$
• Interpretation: any $x$ outside $[-1,1]$ keeps the root positive and the denominator non-zero.

Piecewise-Defined Functions

• Definition: rule changes on sub-intervals of domain.
• Example 1 f(x)=\begin{cases} x^2 & x\ge0,\ 1 & x<0. \end{cases}
  • Graph: right-side parabola for $x\ge0$; constant strip $y=1$ to the left, with an open circle at $x=0$.
• Example 2 (Absolute Value)
  
  |x|=\begin{cases}
  x & x\ge0,\
  -x & x<0.
  \end{cases}
  

Catalogue of Function Types & Typical Domains

• Bullet list of categories the lecture touched on:

1. Linear (degree 1 polynomial)

   • General form $y=mx+b$.
   • Slope-intercept form, $m$ = slope, $b$ = $y$-intercept.
   • Point-slope form $(y-y<em>1)=m(x-x</em>1)$.
   • Domain: $(-\infty,\infty).$

2. Polynomials

   • $P(x)=a<em>nx^n+\dots+a</em>0$, degree $n$.
   • Domain: all real numbers.

3. Power Functions

   • $f(x)=x^p$.
   • Still a polynomial when $p$ is a positive integer; otherwise treated separately.
   • If $p$ is even → symmetric, domain $(-\infty,\infty).$
   • If $p$ is odd → same domain.

4. Algebraic Functions

   • Built from finitely many +, −, ×, ÷, and roots.

5. Rational Functions

   • $f(x)=\dfrac{P(x)}{Q(x)}$ with polynomials $P,Q$.
   • Domain: ${x\mid Q(x)\ne0}$ (all reals minus the roots of $Q$).
   • Example: $\dfrac{x-1}{x-2}$ → domain $(-\infty,2)\cup(2,\infty).$

6. Root Functions ($n$-th roots)

   • $f(x)=x^{1/n}$.
   • $n$ even → domain $[0,\infty).$ (e.g. square root)
   • $n$ odd → domain $(-\infty,\infty).$ (e.g. cube root)

7. Trigonometric Functions

   • $\sin x,\;\cos x,\;\tan x,\dots$
   • Must recall shapes: $\sin$ & $\cos$ oscillate between $\pm1$, $\tan$ has vertical asymptotes at $x=(2k+1)\tfrac\pi2$.

Symmetry of Functions

• Definitions
  • Even: $f(-x)=f(x)$ for all $x$.
    • Graph symmetric about the $y$-axis.
    • Example: $f(x)=x^2$.
  • Odd: $f(-x)=-f(x)$ for all $x$.
    • Graph has rotational symmetry of $180^\circ$ about the origin.
    • Example: $f(x)=x^3$, also $f(x)=\tan x$.

Vertical Line Test (Graph ↔ Function)

• Procedure: Sweep a vertical line across the graph.
• Criterion: If any vertical line intersects the drawing more than once, the graph fails to represent a function (violates uniqueness of output).
• Illustrative non-function: sideways parabola $(x=y^2)$; every vertical line through the “stem” cuts twice.

Additional Concepts Mentioned for Future Lectures

• Mean Value Theorem (to be covered later).
• Periodic functions.
• Increasing / decreasing behaviour.
• Function transformations & combinations (composition, shifts, stretches, etc.).

Practical / Ethical / Real-World Connections

• Height-of-a-person example: mapping “student” → “height”. Demonstrates
  • uniqueness (one height per person),
  • possibility of shared outputs (two people same height),
  • unused codomain values (nobody 7′1″ in the class).
• Student-ID example: injective (one-to-one) function variant where different inputs must also have different outputs (no two students share ID).

Numerical & Algebraic Reminders

• Slope formula: $m=\dfrac{y<em>2-y</em>1}{x<em>2-x</em>1}$ (order matters — be consistent).
• Absolute value identity: $\sqrt{x^2}=|x|$.
• Domain algorithm for rational + root mixtures:
  1. Inside-root expression $\ge0$ (or $>0$ if denominator).
  2. Denominator $\ne0$.
  3. Combine inequalities.
• Interval notation rules:
  • Parentheses $()$ → endpoint excluded.
  • Brackets $[]$ → endpoint included.
  • Union symbol $\cup$ to combine disjoint pieces.