• 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)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