Lecture 1: Functions_Part 1 - Galala University Mathematics I (MAT 111) Fall 2024-2025 PDF

Galala University Fall 2024-2025 Faculty of Science Mathematics I (MAT 111) Department of Mathematics Lecture 1 Functions_Part 1 Dr. Sameh Basha ...

Galala University Fall 2024-2025 Faculty of Science Mathematics I (MAT 111) Department of Mathematics Lecture 1 Functions_Part 1 Dr. Sameh Basha  Standard Sets of Numbers Natural numbers: ℕ = 1,2,3, … , Integer numbers: ℤ = … , −3, −2, −1,0,1,2,3, … 𝑝 Rational numbers: ℚ = { :𝑝 ∈ ℤ 𝑎𝑛𝑑 𝑞 ∈ ℕ} 𝑞 Real numbers: ℝ 2  Notes: Every natural number n in ℕ also will be integer number in ℤ Natural numbers: ℕ = 1,2,3, … , Integer numbers: ℤ = … , −3, −2, −1,0,1,2,3, … Notice that every integer n is also a rational number, since we can write it as the quotient of two integers: n = n/1. 3 4 Note There exist some number not in ℚ and exist in the real line ℝ (i.e., 𝑝 can not represent in the form ) 𝑞 as 𝜋, 2, 5, … Which are called irrational numbers (ℝ\ℚ). ℝ = ℚ ∪ (ℝ\ℚ) 5 Equations and Inequalities Equations and inequalities are both mathematical sentences formed by relating two expressions to each other. In an equation, the two expressions are deemed equal which is shown by the symbol =. Ex: x=y. Ex: 3𝑥 + 5 = 11 In an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, y, x, 𝒃 (ii) if 𝑎 > 𝑏 and b > 𝑐 then 𝐚 > 𝒄 (iii) if 𝑎 > 𝑏 and 𝑐 ∈ ℝ then 𝒂 + 𝒄 > 𝒃 + 𝒄 (iv) if 𝑎 > 𝑏 and 𝑐 ∈ ℝ then 𝒂 − 𝒄 > 𝒃 − 𝒄 (v) if 𝑎 > 𝑏 and 𝑐 > 0 then 𝒂𝒄 > 𝒃𝒄 (vi) if 𝑎 > 𝑏 and 𝑐 < 0 then 𝒂𝒄 < 𝒃𝒄 (vii) if 𝑎 > 𝑏 >0 then 𝟏 < 𝟏 𝒂 𝒃 9 Absolute Value Functions The distance from number 3 to zero is: 3 The distance from number -3 to zero is: 3 The absolute value of a number is the distance from the number to zero. 𝒙, 𝒙 ≥ 𝟎 𝒙 =ቊ −𝒙, 𝒙 < 𝟎 Ex: 𝟑 =𝟑, −𝟑 = 𝟑, −𝟕 = 𝟕 10 Note: 𝒙𝟐 = ±𝒙 𝒙, 𝒙 ≥ 𝟎 𝒙 =ቊ −𝒙, 𝒙 < 𝟎 𝒙𝟐 = 𝒙 11  𝟑 = 𝟑 , −𝟑 = 𝟑 𝟐 𝟐 (𝟑) = 𝟗 , (−𝟑) = 𝟗 12  Inequalities Properties If a,b,c, and d are real numbers then: (viii) if 𝑎2 ≥ 𝑏 2 then 𝑎 ≥ 𝑏 (ix) if 𝑎 > 𝑏 >0 then 𝒂𝟐 > 𝒃𝟐 (x) if 0 > 𝑎 > 𝑏 then 𝒂𝟐 < 𝒃𝟐 (xi) if 𝑎 > 𝑏 >0 then 𝒂 > 𝒃 (xii) if 0 > 𝑎 > 𝑏 then 𝒂 < 𝒃 13 Absolute Value Functions The distance from number 3 to zero is: 3 The distance from number -3 to zero is: 3 The absolute value of a number is the distance from the number to zero. 𝒙, 𝒙 ≥ 𝟎 𝒙 =ቊ −𝒙, 𝒙 < 𝟎 𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈, 𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈 ≥ 𝟎 𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈 = ቊ −(𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈), 𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈 < 𝟎 14  𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈, 𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈 ≥ 𝟎 𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈 = ቊ −(𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈), 𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈 < 𝟎 Ex: 𝟐𝒙 − 𝟑, 𝟐𝒙 − 𝟑 ≥ 𝟎 𝟐𝒙 − 𝟑 = ቊ −(𝟐𝒙 − 𝟑), 𝟐𝒙 − 𝟑 < 𝟎 𝟐𝒙 − 𝟑, 𝟐𝒙 ≥ 𝟑 =ቊ −(𝟐𝒙 − 𝟑), 𝟐𝒙 < 𝟑 𝟑 𝟑 𝟐𝒙 − 𝟑, 𝒙≥ 𝟐𝒙 − 𝟑, 𝒙≥ = 𝟐= 𝟐 𝟑 𝟑 𝟑 − 𝟐𝒙, 𝒙< −(𝟐𝒙 − 𝟑), 𝒙< 𝟐 𝟐 15 Theorem Let 𝑎 > 0, then 𝒙 ≤𝒂 ⟹ −𝒂≤𝒙≤𝒂 𝒙 > 𝒂 ⟹ 𝒙 > 𝒂 𝑶𝑹 𝒙 < −𝒂 16  Ex: Solve the following inequalities: 2𝑥 − 3 < 5 Sol: 2𝑥 − 3 < 5 ⇒ −5 < 2𝑥 − 3 < 5 ⇒ −5 + 3 < 2𝑥 < 5 + 3 ⇒ −2 < 2𝑥 < 8 −2 8 ⇒ 7 Sol: 2𝑥 + 3 > 7 ⇒ 2𝑥 + 3 > 7 OR 2𝑥 + 3 < −7 ⇒ 2𝑥 > 4 OR 2𝑥 < −10 ⇒𝑥>2 OR 𝑥 < −5 S.s= −∞, −5 ∪ 2, ∞ =ℝ − −5,2 20 Theorem Let 𝑎 > 0, then 𝒙 ≤𝒂 ⟹ −𝒂 ≤𝒙≤𝒂 𝒙 > 𝒂 ⟹ 𝒙 > 𝒂 𝑶𝑹 𝒙 < −𝒂 Note: 𝒙𝟐 = 𝒙 (𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈)𝟐 = 𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈 21 Example: Solve the following inequalities: 1- 2𝑥 − 1 < 5 2- 5𝑥 + 2 ≥ 3 3- 2𝑥 + 3 ≤ 4 4- 3 𝑥 2 − 1 < 7 22 Function 23 Function Overview: Functions are fundamental to the study of calculus. In this chapter we review what functions are and how they are visualized as graphs, how they are combined and transformed, and ways they can be classified. Functions are a tool for describing the real world in mathematical terms. A function can be represented by an equation (algebraically), a graph, a numerical table, or a verbal description. 24 Definition 25 Function A function is a rule that assign to each element 𝑥 in a set 𝐷 exactly one element 𝑦 in a set 𝐸. 26 Function A function is a rule that assign to each element 𝑥 in a set 𝐷 exactly one element 𝑦 in a set 𝐸. 27 Function D E D E 1 1 1 1 4 2 2 4 5 3 3 5 8 4 4 9 Not Function Not Function D E D E 1 1 1 1 2 4 4 2 3 5 5 4 4 7 Function Function 28 Function A function is a rule that assign to each element 𝑥 in a set 𝐷 exactly one element 𝑦 in a set 𝐸. 29 Function D E Domain f 1 1 Co-domain f = Df 2 4 3 5 4 8 30 Function D E Domain f 1 1 Co-domain f = Df 4 2 = 𝟏, 𝟒, 𝟓, 𝟖, 𝟏𝟎 5 3 = 𝟏, 𝟐, 𝟑 , 𝟒 8 4 10 Range f = Rf = 𝟏 , 𝟒, 𝟓 , 𝟖 31  Function D E D E Domain f = Df = 𝟏, 𝟐, 𝟑, 𝟒 Domain f = Df = 𝟏, 𝟐, 𝟑, 𝟒 1 1 1 1 4 2 4 2 Co-domain f = 𝟏, 𝟒, 𝟓, 𝟖 Co-domain f = 𝟏, 𝟒, 𝟓, 𝟖, 𝟏𝟐 5 3 5 3 8 4 8 4 Range f = Rf = 𝟏, 𝟒, 𝟓, 𝟖 Range f = Rf = 𝟏, 𝟒, 𝟓, 𝟖 12 Function Function D E D E Domain f = Df = 𝟏, 𝟐, 𝟑, 𝟒 1 1 1 1 4 Co-domain f = 2 2 𝟏, 𝟒, 𝟓 4 5 3 3 5 8 4 4 Range f = Rf = 𝟏, 𝟒, 𝟓 9 Function Not Function 32 Function Co-domain f = 𝒇 𝒙 = 𝒙𝟐 + 𝟐 𝟏, 𝟑, 𝟒, 𝟓, 𝟔, 𝟏𝟏, 𝟏𝟖, 𝟐𝟎 1 3 Domain f = Df = 𝟏, 𝟐, 𝟑, 𝟒 4 Range f = Rf = 𝟑, 𝟔, 𝟏𝟏, 𝟏𝟖 1 5 2 6 3 11 4 18 20 33 Function 𝟏 Generally, 𝒇 𝒙 = Domain f = Df=ℝ − {𝟐} 𝒙−𝟐 −𝟏 Domain f = Df = 𝟎, 𝟏, 𝟑, 𝟒 0 𝟐 1 3 -1 4 1 2 𝟏 𝟐 34 Domain function Generally, Domain the function 𝑓(𝑥) is the set of all possible values of the input 𝑥 that acceptable by the function 𝑓(𝑥). (i.e., that can be processed by the function 𝑓(𝑥) ). 𝐷𝑓 = ℝ − {𝑝𝑜𝑖𝑛𝑡𝑠 𝑡ℎ𝑎𝑡 𝑚𝑎𝑘𝑒 𝑎 𝑝𝑟𝑜𝑏𝑙𝑒𝑚 (𝑁𝑜𝑡 𝑎𝑐𝑐𝑒𝑝𝑡𝑎𝑏𝑙𝑒 𝑏𝑦 𝑓(𝑥))} Points that make the Points that make the denominator = zero value inside the square root is negative 35 Function 𝟏 𝒇 𝒙 = 𝒙−𝟐 Domain f = Df=ℝ − {𝟐} 36 Example: Find 𝐷𝑓 1- 𝑓 𝑥 = 4 − 𝑥 2 2- 𝑓 𝑥 = 𝑥 2 − 9 1 3- f(x)= 𝑥−4 1 4- 𝑓 𝑥 = 𝑥 2 +2 5- 𝑓 𝑥 = 𝑥2 + 2 37 Example: Find 𝐷𝑓 1- 𝑓 𝑥 = 4 − 𝑥 2 Answer: The function 𝑓(𝑥) to be defined should: 4 − 𝑥2 ≥ 0 ⟹ 4 ≥ 𝑥2 ⟹ 𝑥 2≤ 4 ⟹ 𝑥 ≤2 ⟹ −2 ≤ 𝑥 ≤ 2 𝐷𝑓 = −2,2 38 Example: Find 𝐷𝑓 2- 𝑓 𝑥 = 𝑥 2 − 9 Answer: The function 𝑓(𝑥) to be defined should: 𝑥2 − 9 ≥ 0 ⇒ 𝑥 2≥ 9 ⇒ 𝑥 ≥3 ⇒ 𝑥 ≥ 3 𝑂𝑅 𝑥 ≤ −3 𝐷𝑓 = −∞, −3 ∪ 3, ∞ = ℝ − −3,3 39 Example: Find 𝐷𝑓 1 3- f(x)= Draft: 𝑥−4 Answer: 𝒙−𝟒=𝟎 ⇒𝒙=𝟒 𝐷𝑓 = ℝ − 4 40 Example: Find 𝐷𝑓 1 4- 𝑓 𝑥 = Draft: 𝑥 2 +2 𝒙𝟐 + 𝟐 = 𝟎 Answer: ⇒ 𝒙𝟐 = −𝟐 ∵ 𝑥 2 + 2 ≠ 0, ∀𝑥 ∈ ℝ Impossible ∴ 𝐷𝑓 = ℝ 41 Example: Find 𝐷𝑓 5- 𝑓 𝑥 = 𝑥 2 + 2 Answer: 𝐷𝑓 = ℝ, Since it is a polynomial 42 Example: Find Df and Rf: 1- 𝑓 𝑥 = 9 − 𝑥 2 2- 𝑓 𝑥 = 𝑥 2 − 4 3- f(x)=1 − 3 9 − 𝑥 2 4- f(x)=1 − 3 2𝑥 2 − 1 5- 𝑓 𝑥 = 4− 𝑥 1 6- 𝑓 𝑥 = 𝑥−4 1 7- 𝑓 𝑥 = 4+𝑥 2 43 Example: Find Df and Rf: 1- 𝑓 𝑥 = 9 − 𝑥 2 To find 𝑅𝑓 : Let 𝑥 ∈ 𝐷𝑓 Answer: ⟹ −3 ≤ 𝑥 ≤ 3 The function 𝑓(𝑥) to be defined ⟹ 0 ≤ 𝑥2 ≤ 9 should: ⟹ 0 ≥ −𝑥 2 ≥ −9 9 − 𝑥2 ≥ 0 ⟹ −9 ≤ −𝑥 2 ≤ 0 ⟹ 9 ≥ 𝑥2 ⟹ 0 ≤ 9 − 𝑥2 ≤ 9 ⟹ 𝑥 2≤ 9 ⟹ 0 ≤ 9 − 𝑥2 ≤ 3 ⟹ 𝑥 ≤3 ⟹ 0 ≤ 𝑓(𝑥) ≤ 3 ⟹ −3 ≤ 𝑥 ≤ 3 ⟹ 𝑅𝑓 = 0,3 ⟹ 𝐷𝑓 = −3,3 44 Note: If −𝟑 ≤ 𝒙 ≤ 𝟑 ⟹ 𝟗 ≤ 𝒙𝟐 ≤ 𝟗 If −𝟑 ≤ 𝒙 ≤ 𝟑 ⟹ 𝟎 ≤ 𝒙𝟐 ≤ 𝟗 45 Note: If −𝟑 ≤ 𝒙 ≤ 𝟑 ⟹𝟑≤ 𝒙 ≤𝟑 If −𝟑 ≤ 𝒙 ≤ 𝟑 ⟹𝟎≤ 𝒙 ≤𝟑 46 Example: Find Df and Rf: 1- 𝑓 𝑥 = 9 − 𝑥 2 𝐷𝑓 = −3,3 𝑅𝑓 = 0,3 47 Example: Find Df and Rf: 2- 𝑓 𝑥 = 𝑥 2 − 4 To find 𝑅𝑓 : Let 𝑥 ∈ 𝐷𝑓 Answer: ⟹ 𝑥 ∈ ℝ − −2,2 The function 𝑓(𝑥) to be defined ⟹ 𝑥 ≥2 should: ⟹ 𝑥2 ≥ 4 𝑥2 − 4 ≥ 0 ⟹ 𝑥2 − 4 ≥ 0 ⟹ 𝑥2 ≥ 4 ⟹ 𝑥2 − 4 ≥ 0 ⟹ 𝑥 ≥2 ⟹ 𝑓(𝑥) ≥ 0 ⟹ 𝑥 ≥ 2 𝑂𝑅 𝑥 ≤ −2 ⟹ 𝑅𝑓 = 0, ∞ ⟹ 𝐷𝑓 = −∞, −2 ∪ 2, ∞ ⟹ 𝐷𝑓 = ℝ − −2,2 48 Example: Find Df and Rf: 2- 𝑓 𝑥 = 𝑥 2 − 4 𝐷𝑓 = −∞, −2 ∪ 2, ∞ ⟹ 𝐷𝑓 = ℝ − −2,2 𝑅𝑓 = 0, ∞ 49 Example: Find Df and Rf: To find 𝑅𝑓 : Let 𝑥 ∈ 𝐷𝑓 3- f(x)=1 − 3 9 − 𝑥 2 ⟹ −3 ≤ 𝑥 ≤ 3 Answer: ⟹ 0 ≤ 𝑥2 ≤ 9 ⟹ 0 ≥ −𝑥 2 ≥ −9 The function 𝑓(𝑥) to be defined should: ⟹ −9 ≤ −𝑥 2 ≤ 0 ⟹ 0 ≤ 9 − 𝑥2 ≤ 9 9 − 𝑥2 ≥ 0 ⟹ 0 ≤ 9 − 𝑥2 ≤ 3 ⟹ 9 ≥ 𝑥2 ⟹ 0 ≤ 3 9 − 𝑥2 ≤ 9 ⟹ 𝑥2 ≤ 9 ⟹ 0 ≥ −3 9 − 𝑥 2 ≥ −9 ⟹ −9 ≤ −3 9 − 𝑥 2 ≤ 0 ⟹ 𝑥 ≤3 ⟹ −8 ≤ 1 − 3 9 − 𝑥 2 ≤ 1 ⟹ −3 ≤ 𝑥 ≤ 3 ⟹ −8 ≤ 𝑓(𝑥) ≤ 1 ⟹ 𝐷𝑓 = −3,3 ⟹ 𝑅𝑓 = −8,1 50 Example: Find Df and Rf: 3- f(x)=1 − 3 9 − 𝑥 2 𝐷𝑓 = −3,3 𝑅𝑓 = −8,1 51 Example: Find Df and Rf: To find 𝑅𝑓 : Let 𝑥 ∈ 𝐷𝑓 4- f(x)=1 − 3 2𝑥 2 −1 1 1 Answer: ⟹𝑥 ∈ℝ− − , 2 2 The function 𝑓(𝑥) to be defined should: ⟹ 𝑥 ≥ 1 2 2𝑥 2 − 1 ≥ 0 1 ⟹ 𝑥2 ≥ ⟹ 2𝑥 2 ≥1 2 1 ⟹ 2𝑥 2 ≥ 1 ⟹ 𝑥2≥ 2 1 ⟹ 2𝑥 2 − 1 ≥ 0 ⟹ 𝑥 ≥ 2 ⟹ 2𝑥 2 − 1 ≥ 0 1 1 ⟹ 𝑥≥ 𝑂𝑅 𝑥 ≤− ⟹ 2𝑥 2 − 1 ≥ 0 2 2 1 1 ⟹ 𝐷𝑓 = −∞, − ∪ ,∞ ⟹ −3 2𝑥 2 − 1 ≤ 0 2 2 1 1 ⟹ 1 − 3 2𝑥 2 − 1 ≤ 1 ⟹ 𝐷𝑓 = ℝ − − , 2 2 ⟹ 𝑓(𝑥) ≤ 1 ⟹ 𝑅𝑓 = −∞, 1 52 Example: Find Df and Rf: 4- f(x)=1 − 3 2𝑥 2 − 1 1 1 𝐷𝑓 = ℝ − − , 2 2 𝑅𝑓 = −∞, 1 53 Example: Find Df and Rf: 5- 𝑓 𝑥 = 4 − 𝑥 To find 𝑅𝑓 : Let 𝑥 ∈ 𝐷𝑓 Answer: ⟹ −4 ≤ 𝑥 ≤ 4 The function 𝑓(𝑥) to be defined ⟹0≤ 𝑥 ≤4 should: ⟹ 0 ≥ − 𝑥 ≥ −4 ⟹4− 𝑥 ≥0 ⟹ −4 ≤ − 𝑥 ≤ 0 ⟹4≥ 𝑥 ⟹0≤4− 𝑥 ≤4 ⟹ 𝑥 ≤4 ⟹0≤ 4− 𝑥 ≤2 ⟹ −4 ≤ 𝑥 ≤ 4 ⟹ 0 ≤ 𝑓(𝑥) ≤ 2 ⟹ 𝐷𝑓 = −4,4 ⟹ 𝑅𝑓 = 0,2 54 Example: Find Df and Rf: 5- 𝑓 𝑥 = 4 − 𝑥 𝐷𝑓 = −4,4 𝑅𝑓 = 0,2 55 Example: Find Df and Rf: 1 6- 𝑓 𝑥 = To find 𝑅𝑓 : Let 𝑥 ∈ 𝐷𝑓 𝑥−4 Answer: Note: ⟹𝑥 ∈ℝ− 4 ⟹ 𝐷𝑓 = ℝ − 4 𝑥−4=0 ⟹ 𝑥 ∈ −∞, 4 ∪ 4, ∞ 𝑥=4 𝑥 ∈ −∞, 4 𝑥 ∈ 4, ∞ ⟹ 𝐷𝑓 = −∞, 4 ∪ 4, ∞ ⟹ −∞ < 𝑥 < 4 ⟹4 𝒂 ⟹ 𝒙 > 𝒂 𝑶𝑹 𝒙 < −𝒂 Note: 𝒙𝟐 = 𝒙 (𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈)𝟐 = 𝒂𝒏𝒚𝒕𝒉𝒊𝒏𝒈 60  The graph of a function is a curve in the xy-plane. But the question arises: Which curves in the xy-plane are graphs of functions? 61 The Vertical Line Test A curve in the xy-plane is the graph of a function of x If and only if No vertical line intersects the curve more than once. 62 The Vertical Line Test A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once. 63 Four Ways to Represent a Function There are four possible ways to represent a function: verbally (by a description in words) numerically (by a table of values) visually (by a graph) algebraically (by an explicit formula) 64 Four Ways to Represent a Function A. Verbally: The cost C of mailing an envelope depends on its weight w. Although there is no simple formula that connects w and C, the post offie has a rule for determining C when w is known. Numerically: The human population of the world P depends on the time t. The table gives estimates of the world population P(t) at time t, for certain years. For instance, P(1950)= 2,560,000,000 Graphically: The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t. Algebrically: The area A of a circle depends on the radius 2 r of the circle. The rule that connects r and A is given by the equation 𝐴 = 𝜋𝑟. 65 piecewise Defined Functions The functions in the following four examples are defied by different formulas in different parts of their domains. Such functions are called piecewise defied functions. 66