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Functions

  • One-to-one Functions: Each x value maps to one distinct y value
    • E.g. f(x) = 3x – 1
  • Many-to-one Functions: There are some f(x) values which are generated by more than one x value
    • E.g. f(x) = x² – 2x + 3
    • Domain = x values
    • Range = y values
  • Notation: f(x) can also be written as f: x →
  • To find range:
    • Complete the square
      • x² – 2x + 3 ⇒ (x – 1)² + 2
    • Work out the min/max point
      • Minimum point = (1, 2)
    • ∴ All y values are greater than or equal to 2. f(x) ≥ 2
  • One-to-many functions do not exist
  • Domain of g(x) = Range of g-1(x)
  • Solving functions::
    • f(2): substitute x = 2 and solve for f(x)
    • fg(x): substitute x = g(x)
    • f-1(x): let y = f(x) and x the subject
  • Transformation of graphs:
    • f(-x): reflection in the y-axis
    • -f(x): reflection in the x-axis
    • f(x) + a: translation of a units parallel to y-axis
    • f(x + a): translation of – a units parallel to x-axis
    • f(ax): stretch, scale factor 1/a parallel to x-axis
    • af(x): stretch, scale factor a parallel to y-axis
  • Modulus Function:
    • Denoted by |f(x)|
    • Modulus of a number is its absolute value
    • Never goes below x-axis
    • Takes a negative graph into positive by reflecting the negative part into the x-axis
  • Solving modulus function:
    • Sketch graphs and find the points of intersection
    • Square the equation and solve the quadratic
  • Relationship of a function and its inverse:
    • The graph of the inverse of a function is the reflection of a graph of the function in y = x
Quadratic Functions
  • To sketch y = ax² + bx + c when a ≠ 0
    • Use the turning point:
      • Express y = ax² + bx + c, as y = a(x – h)² + k, by completing the square
      • x² + nx ⇔ (x + n/2)² – (n/2
      • a(x + n)² + k
      • where the vertex is (-n, k)
    • a > 0: u-shaped, therefore it is the minimum point
    • a < 0: n-shaped, therefore it is the maximum point
  • Find the x-intercept
    • Factorize or use the quadratic formula
  • Get the type of roots by calculating discriminant b² – 4ac
    • If b² – 4ac = 0, real and equal roots
    • If b² – 4ac > 0, real and distinct roots
    • If b² – 4ac < 0, no real roots
  • Intersections of a line and a curve: If the simultaneous equations of the line and curve leads to a simultaneous equation, then:
    • If b² – 4ac = 0, the line is tangent to the curve
    • If b² – 4ac > 0, the line meets the curve at two points
    • If b² – 4ac < 0, the line does not meet the curve
  • Quadratic inequality:
    • (x – d)(x – β) < 0 ⇒ d < x < β
    • (x – d)(x – β) > 0 ⇒ x < d or x > β
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