Functions and Relations

Functions and Relations

Relations

  • Show a relationship between some x and y values.
  • Described by a rule, a graph, or co-ordinate points (x,y)(x, y).
    • xx value: independent variable.
    • yy value: dependent variable.
  • Can be continuous or discrete.
Types of relations:
  • One-to-one: One unique yy value for every one xx.
  • Many-to-one: More than one xx-value for one yy-value.
  • One-to-many: More than one yy-value for one xx-value.
  • Many-to-many: More than one xx-value for any yy-value.

Functions

  • A relation which is one-to-one or many-to-one.
  • Vertical Line Test: Used to classify functions from a graph.
    • If a vertical line intersects the graph more than once, it is not a function.

Interval Notation

  • Representing an interval using end values and indicating inclusion/exclusion.
    • (),<,>,◯( ), <, > , \bigcirc: Excludes the end values.
    • [ ], \geq, \leq, \Circle: Includes the end values.

Domain and Range

  • Domain: The set of independent (x)(x) values.
  • Range: The set of dependent (y)(y) values.
  • Expressed using interval notation.

Function Notation

  • Expressing 'y' as f(x)f(x), g(x)g(x), etc.
  • E.g., to substitute x=2x=2 into y=x2y = x^2, write f(2)f(2).

Piece-wise Functions

  • Uses different rules for different sections of the domain (x values).
Constructing a continuous piece-wise graph
  1. Solve the equations simultaneously to determine the point of intersection.
  2. Sketch the piece-wise linear graph.

Circles

Equation:
  • Centre is (0,0)(0, 0): x2+y2=r2x^2 + y^2 = r^2 where rr is the radius.
  • Centre is (h,k)(h, k): (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 where rr is the radius.
Steps for Sketching:
  1. Identify the circle's center and radius.
  2. Find x and y intercepts (if they exist).
  3. Graph the information and sketch the circle.
Semicircles
  • Rearrange the circle formula to make y the subject.

Relation y2=xy^2 = x (Sideways Parabola)

  • Cannot be a function.
  • Key features:
    • Opens to the right for y2=xy^2 = x or to the left for y2=−xy^2 = -x.
    • Vertex at (0,0)(0,0).
    • Axis of symmetry is horizontal with equation y=0y=0 (the x-axis).

Square Root Function y=ax−h+ky = a\sqrt{x-h} + k

  • Vertex is (h,k)(h, k), axis of symmetry is y=ky = k
  • If aa is positive, range is [k,∞)[k, \infty); end point is a minimum (