copyright by Jordan Lau
Cambridge (CIE) IGCSE Additional Maths – Function
Notation
⚫ an expression of a series of operations in
terms of input(s)
⚫ donate by f(x), g(x) or other alphabets
e.g. f(x) = 3x+1
where x is input and f(x) is output
Question Style
⚫ state the domain and range of a given
function
⚫ e.g. 𝑓(𝑥) = 𝑥 2 + 1
◼ domain: (−∞, +∞)
◼ range: [1, +∞)
Definition
Caution
⚫ many (input) to one (output) or one to one ⚫ zero cannot be denominator, it will affect the
⚫ it cannot be one to many or many to many
domain
Graphical Check
⚫ f(x) is a function ⟺ only one intersection
between the graph of f(x) and any vertical
line
Domain and Range
⚫
⚫
Domain: set of values for input
Range: set of values for output
Presentation of Domain / Range
⚫ (a, b) ≡ 𝑎 < 𝑥 < 𝑏
⚫ (c, d] ≡ 𝑐 < 𝑥 ≤ 𝑑
⚫ (−∞, +∞) ≡ 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙
⚫ (−∞, 3) 𝑜𝑟 (3, +∞) ≡ 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑒𝑥𝑐𝑒𝑝𝑡 3
⚫ (0, +∞) ≡ 𝑎𝑙𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑟𝑒𝑎𝑙 ≡ 𝑥 ∈ ℝ+
⚫
1
𝑒. 𝑔. 𝑓 (𝑥) = 1−𝑥
◼
◼
domain: (−∞, 1) 𝑜𝑟 (1, +∞)
range: (−∞, 0) 𝑜𝑟 (0, +∞)
Inverse Function & Notation
⚫
Does exact opposite of the function it came
from.
It is denoted by 𝑓 −1 (𝑥)
⚫
e.g. f(x) = 2x; 𝑓 −1 (𝑥) = 2
⚫
inverse function exist ⟺ 𝑓(𝑥)𝑖𝑠 𝑜𝑛𝑒 𝑡𝑜 𝑜𝑛𝑒
⚫
𝑥
How to find inverse function; 𝑓(𝑥) = 2𝑥 + 1
1. Replace f(x) by y; 𝑦 = 2𝑥 + 1
2. Swap x and y; 𝑥 = 2𝑦 + 1
𝑥−1
3.
Making y as subject; 𝑦 =
4.
Apply correct notation; 𝑓 −1 (𝑥) =
2
𝑥−1
2
Properties of Inverse Function
⚫ Swapping input and output of f(x)
◼ 𝑓(3) = 10; 𝑓 −1 (10) = 3
⚫ Swapping domain and range of f(x)
⚫
⚫
1
𝑥−1
◼
𝑓 (𝑥) = 1−𝑥 ; 𝑓 −1 (𝑥) =
◼
◼
𝑓(𝑥)𝑑𝑜𝑚𝑎𝑖𝑛: (−∞, 1) 𝑜𝑟 (1, +∞)
𝑓(𝑥)𝑟𝑎𝑛𝑔𝑒: (−∞, 0) 𝑜𝑟 (0, +∞)
𝑥
◼ 𝑓 −1 (𝑥) 𝑑𝑜𝑚𝑎𝑖𝑛: (−∞, 0) 𝑜𝑟 (0, +∞)
◼ 𝑓 −1 (𝑥) 𝑟𝑎𝑛𝑔𝑒: (−∞, 1) 𝑜𝑟 (1, +∞)
Inverse of inverse function = x
◼ 𝑓 −1 (𝑓 −1 (𝑥)) = 𝑓(𝑥)
𝑓(𝑓 −1 (𝑥)) = 𝑓 −1 (𝑓(𝑥)) = 𝑥
copyright by Jordan Lau
Graph of Inverse Function
⚫ Reflection of the graph f(x) with
⚫
◼
respect to y=x
𝑓(𝑥) = 4 − 4𝑥, (𝑏𝑙𝑎𝑐𝑘 𝑜𝑛𝑒)
𝐷: (0, +∞); 𝑅: (−∞, 4)
⚫
𝑓 −1 (𝑥) =
◼
𝐷: (−∞, 4); 𝑅: (0, +∞)
4−𝑥
4
, (𝑔𝑟𝑒𝑒𝑛 𝑜𝑛𝑒)
Composite Function & Notation
⚫ One function is applied after another
⚫
⚫
⚫
function
𝑖𝑛𝑝𝑢𝑡 → 𝑔( )𝑜𝑢𝑡𝑝𝑢𝑡 = 𝑓( )𝑖𝑛𝑝𝑢𝑡 → 𝑜𝑢𝑡𝑝𝑢𝑡
𝑓𝑔(𝑥) 𝑜𝑟 𝑓°𝑔(𝑥) 𝑜𝑟 𝑓(𝑔(𝑥))
𝑖𝑓 𝑓 𝑎𝑛𝑑 𝑔 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒, 𝑖𝑡 𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑎𝑠 𝑓 2 (𝑥)
Caution! It is different from (𝑓(𝑥))2
Properties of Composite Function
⚫ 𝑓°𝑔(𝑥) 𝑚𝑎𝑦 𝑛𝑜𝑡 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑔°𝑓(𝑥)
Caution: for 𝑓°𝑔(𝑥)
Its domain is the domain of g(x)
Its range will be the range of f(x)
But need to verify whether range of g(x) satisfy
domain of f(x)
⚫