IGCSE Additional Mathematics
Functions
Domain and Range
Types of Functions
Format
Modifications to existing functions
Polynomials
Many-to-One Functions
Domain
Range of Inputs
Range
Range of Possible Outputs :
f:x |-> x
f(x) = x
Quadratic Functions
One-to-One Functions
f(x) = cos x where 0 <= x <= 180
f(x) = n^x where n>=0
f(x) = n^(2x-1) where x is an integer
Inverse Functions
Modulus Function
Aka Absolute Value or Magnitude of the Value
Denoted by ||
|-21| = 21
Cubic Functions ( That have 2 / 3 unique roots )
Linear Functions
f(x) = x^2 - 1
x > 0
f(x) > -1
Composite Functions
f(x)=2x , g(x)=3x | fg(x)= 2(3x) = 6x
f(x) = 2x - 1 | x = 2y - 1 | 2y = x + 1 | y = ( x + 1 )/2 | f'(x) = ( x + 1 )/2
f(x) = 2x - 1
Minimum/Maximum Points
Methods to Solve Quadratic Functions
Quadratic Inequalities
Finding the number of roots in a quadratic function
Discriminant
Uses
Finding no of roots
If b^2 - 4ac < 0, no roots
Factorisation
Completing the Square
Quadratic Equation
(-b +- √(-b^2 - 4ac))/2a
x^2 -2x + 1 = 0 | ( x - 1 )^2 = 0 | x = 1
Types of Quadratic Functions
x^2 + 7x -17 = 0 | ( x^2 + 7x + 49/4 ) - 17 = 49/4 | ( x + 7/2 )^2 = 117/4 | x = +- √(117/4) - 7/2 | x = 1.91 , -8.91
Complete
Missing constant?
Missing middle term
x^2 - 1 = 0 | x^2 = 1 | x = +-1
x^2 + 1 = 0 | x^2 = -1 | x = n/a
2x^2 + 2x = 0 | 2x ( x + 1 ) = 0 | x = 0 , -1
f(x) = x^2 - x + 1
x^2 - x + 1/4 > 0 | ( x - 1/2 )^2 > 0 | x = 1/2
X intercepts at ( 0 , 2 ) , ( 0 , 3 ) | Minimum/Maximum point's y=2.5 as ( 3 + 2 )/2 = 2.5
Minimum if x^2 is positive | Maximum if x^2 is negative
If b^2 - 4ac = 0, one recurring root
If b^2 - 4ac > 0, 2 unique roots
Finding unknown value within quadratic functions
?
x^4 - x^2 +1/2 = 0 | u=x^2 and u^2 - u + 1/2 = 0
Modulus Functions
Modulus is denoted by ||, aka Absolute Value or Magnitude of a Value
| x + 1 | = 2 // x + 1 = 2 and x + 1 = -2
Modulus Inequalities
| x + 1 | > 2 // x + 1 > 2 and x + 1 < -2
If both sides are constrained by a modulus, the only the answer when bot sides are positive, and one side is negative should be considered