**One-to-one Functions**: Each x value maps to one distinct y value
**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
- ∴ 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