Unit 1

1.1 What Is and Isn’t a Function?

1. Vertical Line Test:

   • A graph is a function if every vertical line intersects it at most once. Fails if any vertical line intersects more than once.

2. Rules for Mappings:

   • Function: Each xx (input) maps to exactly one yy (output).
   • Not a Function: An xx maps to multiple yy values (e.g., vertical overlaps).
   • Repeated xx values with identical yy values do not violate the rule.

3. Set Representation Example:
   $(1,2),(1,2),(3,5)\{(1,2),(1,2),(3,5)\}$:

   • Function: Each input has a single consistent output. Duplicates in sets are ignored.
   • Not a Function: If it were $(1,2),(1,3)\{(1,2),(1,3)\}$, input 11 maps to multiple outputs.

4. Graph Examples:

   • Function: Quadratic $(y=x2y=x^2)$, linear (y=mx+by = mx+b).
   • Not a Function: Circle $(x2+y2=r2x^2+y^2=r^2)$, sideways parabola $(x=y2x=y^2)$.

1.2 Function Notation

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To Review

• Circles: $x2+y2=r2x^2+y^2=r^2$: Analyze intersections to test functionality.
• Conversions: Between general forms and standard forms using completing the square or isolating variables.
• Perfect Squares:
  • Expansion: $(a+b)2=a2+2ab+b2(a+b{)}^2=a^2+2ab+b^2$.
  • Recognition: $a2-2ab+b2=(a-b)2a^2-2ab+b^2=(a-b{)}^2.$

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