9758 H2 Mathematics Notes

1 - Pure Math 1 - Functions & Graphs A - Functions 1 - Notations A - Set A. ∈ / ∉ ➢ {Element} is / is not an element of {set} ➢ 2 ∈ / ∉ A : The real number 2 is / is not an element of the set A B. ℝ / ℝ⁺ / ℝ⁻ ➢ The set of all / positive / negative real numbers, such as fractions, decimals, and integers, etc… C. ℤ / ℤ⁺ / ℤ⁻ ➢ The set of all / positive / negative integers ➢ ℤ / ℤ⁺ / ℤ⁻ = {-3,-2,-1,0,1,2,3,...} / {1,2,3,...} / {...,-3,-2,-1} ➢ 0 ∉ ℤ⁺ / ℤ⁻ D. ⊆ a. {set_A} is a subset of {set_B} b. Everything from set A is inside of set B c. 𝑥 = { 1 , 2 , 3 }d. 𝑦 = { 1 , 2 , 3 , 4 , 5 }e. Since contains all of the elements from , is a subset of 𝑦 𝑥 𝑥 𝑦 B - Interval ● An alternative way of showing a set that’s bounded by a range of values ● Type 1 : Bounded ➢ Bracket : Rounded () ➢ Is exclusive of the intervals’ values ➢ ( 𝑎 , 𝑏 ) = { 𝑆𝐸𝑇 𝑁𝑂𝑇𝐴𝑇𝐼𝑂𝑁 : 𝑎 < 𝑏 }➢ Example : ( 0 , 9 ) = { 𝑥 ∈ 𝑅 : 0 < 9 } ➢ The set notation is bounded by the intervals 0 and 9, excluding 0 and 9 themselves 𝑥 ∈ 𝑅 ● Type 2 : Unbounded ➢ Bracket : Square [] ➢ Is inclusive of the intervals’ values ➢ [ 𝑎 , 𝑏 ] = { 𝑆𝐸𝑇 𝑁𝑂𝑇𝐴𝑇𝐼𝑂𝑁 : 𝑎 ≤ 𝑥 ≤ 𝑏 }➢ Example : [ 0 , 9 ] = { 𝑥 ∈ 𝑅 : 0 ≤ 𝑥 ≤ 9 } ➢ The set notation is bounded by the intervals 0 and 9, including 0 and 9 themselves 𝑥 ∈ 𝑅

● Type 3 : Hybrid ➢ Bracket : Both (] [) ➢ Is both inclusive and exclusive of the intervals’ values ➢ ( 𝑎 , 𝑏 ] = { 𝑆𝐸𝑇 𝑁𝑂𝑇𝐴𝑇𝐼𝑂𝑁 : 𝑎 < 𝑥 ≤ 𝑏 }➢ [ 𝑎 , 𝑏 ) = { 𝑆𝐸𝑇 𝑁𝑂𝑇𝐴𝑇𝐼𝑂𝑁 : 𝑎 ≤ 𝑥 < 𝑏 } ➢ Example : ( 0 , 9 ] = { 𝑥 ∈ 𝑅 : 0 < 𝑥 ≤ 𝑏 } ➢ The set notation is bounded by the intervals 0 and 9, excluding 0 and including 9 𝑥 ∈ 𝑅➢ Example : [ 0 , 9 ) = { 𝑥 ∈ 𝑅 : 0 ≤ 𝑥 < 𝑏 } ➢ The set notation is bounded by the intervals 0 and 9, including 0 and excluding 9𝑥 ∈ 𝑅

2 - Types A - Generic ➢ A mapping of exactly 1 value from a set of inputs to exactly 1 value from a set of outputs ➢ For example, the equation will map 2 to 4 , 4 to 8, etc… 𝑓 ( 𝑥 ) = 2 𝑥● It’s defined by its rule and domain 1. Visuals ○ 𝑓 : 𝑖𝑛𝑝𝑢𝑡 ↦ { 𝑜𝑢𝑡𝑝𝑢𝑡 } 𝑓𝑜𝑟 { 𝑑𝑜𝑚𝑎𝑖𝑛 }■ Example : 𝑓 : 𝑥 ↦ 2 𝑥 𝑓𝑜𝑟 𝑥 ≥ 0 ■ The output is the rule of the function 2 𝑥 𝑓 ○ 𝑓 ( 𝑥 ) = { 𝑜𝑢𝑡𝑝𝑢𝑡 } 𝑓𝑜𝑟 { 𝑑𝑜𝑚𝑎𝑖𝑛 }■ Example : 𝑓 ( 𝑥 ) = 2 𝑥 𝑓𝑜𝑟 𝑥 ≥ 0 ○ 𝑦 = { 𝑜𝑢𝑡𝑝𝑢𝑡 } 𝑓𝑜𝑟 { 𝑑𝑜𝑚𝑎𝑖𝑛 }■ Example : 𝑦 = 2 𝑥 𝑓𝑜𝑟 𝑥 ≥ 0 ★ The domain is not compulsory to define a function 2. Domain & Rule ○ Domain ■ Shown as 𝐷 𝑓 ( 𝑥 ) ■ The set of inputs ■ Graphically it’s the x-axis ■ Determines how the graph will look like horizontally ■ Makes the function unique ○ Rule