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