Functions: Mapping, Domain/Codomain, and Independent Variable

Functions and Their Mapping

• Core Idea: A function takes an input and gives exactly one output according to a rule.

• Key Parts (Notation):

  • Domain (D): The set of all possible inputs.

  • Codomain (R): The set of all possible outputs.

  • Notation: $f: D \to R$

  • For every input $x$ in D, the output is $f(x)$ in R.

• The "One Output" Rule (Formal):

  • For every input $x$ in the Domain D, there is one and only one output $y$ in the Codomain R, such that $y=f(x)$.

  • Or: $\forall x\in D,\exists!\, y\in R\text{ such that } y=f(x)$.

  • This means: Each input must map to exactly one output.

• Mapping Rules:

  • An input can't have two different outputs.

  • Different inputs CAN have the same output. (e.g., $f(x<em>1) = f(x</em>2)$ even if $x<em>1 \neq x</em>2$). Example: $x^2$; both 2 and -2 give 4.

  • Not all possible outputs in the Codomain R have to be used by an input.

• Ways to Define a Function:

  • Algebraic (Formula): Using an equation like $y=f(x)=x^2$.

  • Numeric (Table/Rule): A list of inputs and their outputs (like a pricing rule based on weight).

• Variables:

  • x: The independent variable (the input you choose from the domain).

  • f(x) (or y): The dependent variable (the output, which depends on x).

• Graphing: Plot points $(x, f(x))$ for all $x$ in the domain D. This shows how outputs change with inputs.

• Real-World Example: Grades and Registrar

  • Domain: Students.

  • Codomain: Possible Grades (A, B, C).

  • Rule: Each student gets exactly one grade.

  • Observations:

  • Every student gets one grade (no student gets two, no student gets none).

  • Different students can get the same grade (e.g., multiple students get an A).

• Key Takeaways:

  • Deterministic: Functions are predictable; the same input always gives the same unique output.

  • Flexible: Can be defined by formulas, tables, or descriptive rules.

  • Domain Matters: A function only works for inputs inside its defined domain D.

• Summary of Notation/Formulas to Remember:

  • Function mapping: $f: D \to R, \quad f(x) \in R\quad (x\in D)$

  • Uniqueness: $\forall x\in D,\exists!\, y\in R \text{ with } y=f(x)$

  • Many-to-one is okay: $\exists x<em>1, x</em>2\in D,\; x<em>1\neq x</em>2 \text{ such that } f(x<em>1)=f(x</em>2)$

  • Example: $y=f(x)=x^2$

  • Graphing: Plot $(x, f(x))$ points.

Practice Problems

1. Identify Domain/Codomain/Function Status: For each given relationship, state if it is a function. If it is, identify a suitable domain and codomain.
   a. A rule that assigns to each person in a classroom their current age.
   b. A rule that assigns to each age the names of all people in the classroom who have that age.
   c. The formula $y = \frac{1}{x-3}$. (Consider all real numbers.)
   d. The set of ordered pairs $(1, 5), (2, 7), (1, 8), (3, 6)$.

2. Formal Notation: Write the formal notation for a function $g$ that maps integers ($\mathbb{Z}$) to integers ($\mathbb{Z}$) such that $g(x) = x^3$.

3. Function Properties: Consider the function $h: \mathbb{R} \to \mathbb{R}$ defined by $h(x) = |x|$.
   a. Can two different inputs map to the same output? Provide an example.
   b. Can a single input map to two different outputs? Explain why or why not, referencing the definition of a function.

4. Real-World Application: A vending machine dispenses one specific snack item when a corresponding button is pressed. If the set of buttons is the domain and the set of snack items is the codomain, is this