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 ■ Shown as 𝑅 𝑓 ( 𝑥 ) ■ The set of outputs ■ Graphically it’s the y-axis ■ Determines how the graph will look like vertically ■ Dependent on its function’s domain ★ Functions with the same rule but different domains are considered different functions ○ A function is a restriction of function if 𝑥 𝑦 𝐷 𝑥 ∈ 𝐷 𝑦

B - Piecewise ● It’s a function which is defined by multiple sub-functions, whereby each sub-function only applies to a specific interval of the function’s domain when is in and when is in 𝑓 ( 𝑥 ) = 𝑔 ( 𝑥 ) 𝑥 ( 𝑎 , 𝑏 ) 𝑓 ( 𝑥 ) = ℎ ( 𝑥 ) 𝑥 ( 𝑤 , 𝑧 ) C - One-To-One ● It’s a function whereby exactly 1 value from the range corresponds to exactly 1 value from the domain ● Horizontal Line Test ○ To test if the given function is one-to-one ○ Pass ■ If any horizontal line cuts the graph at most once , 𝑦 = 𝑘 , ( 𝑘 ∈ ℝ ) 𝑦 = 𝑓 ( 𝑥 )then the function is a one-to-one function 𝑦 = 𝑓 ( 𝑥 )○ Fail ■ If any horizontal line cuts the graph more than once , 𝑦 = 𝑘 , ( 𝑘 ∈ ℝ ) 𝑦 = 𝑓 ( 𝑥 )then the function is not a one-to-one function 𝑦 = 𝑓 ( 𝑥 ) D - Inverse ● It’s a function that reverses the domain and range of another function 𝑔 ( 𝑥 ) 𝑓 ( 𝑥 )● Written as 𝑓 − 1 ( 𝑥 )● Hence, and 𝐷 𝑓 − 1 ( 𝑥 ) = 𝑅 𝑓 ( 𝑥 ) 𝑅 𝑓 − 1 ( 𝑥 ) = 𝐷 𝑓 ( 𝑥 ) ★ In order for a function to have an inverse counterpart 1. It must be a one-to-one function ● To create a function’s inverse 1. Make the variable from the RHS the subject 1. 𝑓 ( 𝑥 ) = 3 𝑥 + 72. 𝑦 = 3 𝑥 + 73. 𝑥 = 𝑦 − 7 3 4. 𝑦 = 𝑥 − 7 3 5. 𝑓 − 1 ( 𝑥 ) = 𝑥 − 7 3 ★ is a reflection of in the line . Hence 𝑓 − 1 ( 𝑥 ) 𝑓 ( 𝑥 ) 𝑓 ( 𝑥 ) = 𝑥1. They will intersect there 2. 𝑓 ( 𝑎 , 𝑏 ) = 𝑓 − 1 ( 𝑏 , 𝑎 )2. If has a vertical asymptote , will have a horizontal asymptote 𝑓 ( 𝑥 ) 𝑥 = 𝑎 𝑓 − 1 ( 𝑥 ) 𝑦 = 𝑎3. If has a horizontal asymptote , will have a vertical asymptote 𝑓 ( 𝑥 ) 𝑦 = 𝑎 𝑓 − 1 ( 𝑥 ) 𝑥 = 𝑎

E - Composite ● It’s a function that contains another function within it (Nested Function) 𝑓𝑔 ( 𝑥 ) = 𝑓 ( 𝑔 ( 𝑥 ) ) 𝑇ℎ𝑒 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑓𝑔 / 𝑓 ∘ 𝑔 ℎ𝑎𝑠 𝑡ℎ𝑒 𝑖𝑛𝑛𝑒𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑔 ( 𝑥 ) 𝑖𝑛𝑠𝑖𝑑𝑒 𝑡ℎ𝑒 𝑜𝑢𝑡𝑒𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑓 ( 𝑥 ) 1. For a composite function to exist 𝑓 ( 𝑔 ( 𝑥 ) )★ 𝑅 𝑔 ⊆ 𝐷 𝑓 ★ 𝐷 𝑓𝑔 = 𝐷 𝑔 1. To find 𝑅 𝑓𝑔 ○ Method 1 1. Sketch the graph of based on 𝑦 = 𝑓𝑔 ( 𝑥 ) 𝐷 𝑓𝑔 2. Obtain the range from the graph 2. Method 2 1. Find 𝑅 𝑔 2. Sketch 𝑦 = 𝑓 ( 𝑥 ) 𝑓𝑜𝑟 𝑅 𝑔 3. will be the corresponding set of values for y 𝑅 𝑓𝑔 ★ 𝑔𝑓 ≠ 𝑓𝑔➢ The order matters ★ 𝑓 − 1 ( 𝑓 ( 𝑥 ) ) = 𝑥 𝑓𝑜𝑟 𝑥 ∈ 𝐷 𝑓 ( 𝑥 ) ➢ This is the identity function 𝑦 = 𝑥 𝑓𝑜𝑟 𝑥 ∈ 𝐷 𝑓 ( 𝑥 ) ★ 𝑓 𝑓 − 1 ( 𝑥 ) = 𝑥 𝑓𝑜𝑟 𝑥 ∈ 𝐷 𝑓 − 1 ( 𝑥 ) ➢ This is the identity function 𝑦 = 𝑥 𝑓𝑜𝑟 𝑥 ∈ 𝐷 𝑓 − 1 ( 𝑥 )

B - Graphs & Transformations 1 - Graph Properties A - Features 1 - Axes - Intercept ● : Value of when 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 𝑥 𝑦 = 0● : Value of when 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 𝑦 𝑥 = 0★ Not all graphs will intercept the axes 2- Stationary Points ● Points on the graph when the gradient 𝑚 = 0● Type 1 : Turning Point (Max / Min) ● Type 2 : Stationary Point Of Inflexion 3 - Asymptotes ● A straight line that a curve approaches as either one of its variables 𝑥 / 𝑦 → ∞● The function will never intercept this line ★ Type 1 : Vertical ➢ is a vertical asymptote of the curve if as 𝑥 = 𝑎 𝑦 = 𝑓 ( 𝑥 ) 𝑦 → ∞ 𝑥 → 𝑎★ Type 2 : Horizontal ➢ is a horizontal asymptote of the curve if as 𝑦 = 𝑎 𝑦 = 𝑓 ( 𝑥 ) 𝑦 → 𝑎 𝑥 → ∞★ Type 3 : Oblique ➢ 𝑦 = 𝑚𝑥 + 𝑐 , 𝑚 ≠ 0 4 - Axes Of Symmetry ● A straight line that divides a graph into 2 equal halves that are reflections of each other, such as ★ Vertical : 𝑥 = 𝑎★ Horizontal : 𝑦 = 𝑏★ Oblique : 𝑦 = 𝑚𝑥 + 𝑐

B - Types 1 - Linear ➢ It’s a straight line ➢ 𝑦 = 𝑚𝑥 + 𝑐 , 𝑚 = 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 & 𝑐 = 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡➢ 2 distinct linear lines are parallel if 𝑚 1 = 𝑚 2 ➢ 2 distinct linear lines are perpendicular to each other if 𝑚 1 𝑚 2 = − 1 2 - Quadratic ➢ It’s a curve with a degree of 2 ➢ 𝑦 = 𝑎 𝑥 2 + 𝑏𝑥 + 𝑐 , { 𝑎 , 𝑏 , 𝑐 } ∈ ℝ , 𝑎 ≠ 0 ■ If , the graph has a minimum point 𝑎 > 0■ If , the graph has a maximum point 𝑎 < 0 ★ To find the root, use the quadratic formula ■ 𝑥 = − 𝑏 ± 𝑏 2 − 4 𝑎𝑐2 𝑎 ★ To determine the roots’ nature, use the discriminant 𝑏 2 − 4 𝑎𝑐■ If , the function has 2 real distinct roots 𝑏 2 − 4 𝑎𝑐 > 0■ If , the function has 2 real equal roots 𝑏 2 − 4 𝑎𝑐 = 0■ If , the function has 0 real root 𝑏 2 − 4 𝑎𝑐 < 0 ★ In