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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
• 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