- 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