Math 103 Lecture 1 PDF

Math 103 Lecture 1 Dr. Mustafa El-Agamy Dr. Mohammad Yasin  Numbers ▪ Natural numbers: ℕ = {1, 2, 3, 4, … } ▪ Whole numbers: ℕ ∗ = {0, 1, 2, 3, 4, … } ▪ Integer numbers: ℤ = { … , −3, −2, −1, 0, 1, 2, 3, … }...

Math 103 Lecture 1 Dr. Mustafa El-Agamy Dr. Mohammad Yasin  Numbers ▪ Natural numbers: ℕ = {1, 2, 3, 4, … } ▪ Whole numbers: ℕ ∗ = {0, 1, 2, 3, 4, … } ▪ Integer numbers: ℤ = { … , −3, −2, −1, 0, 1, 2, 3, … } 1 4 −10 ▪ Rational numbers: ℚ = {𝒏/𝒎; 𝑛, 𝑚 ∈ ℤ; 𝑚 ≠ 0} 2 3 ത Examples: = 0.5, = 1. 3, 5 = −2 ▪ Irrational numbers: ℚ′ = {ⅇ, 𝜋, 2 , … } ▪ Real numbers: ℝ = ℚ ⋃ ℚ′ = −∞, +∞ ℕ, ℤ and ℚ contain discrete data values, while ℝ contains continuous data values. Venn Diagram of Number System  Functions 𝒚=𝒇 𝒙  Vertical Line Test A function ƒ can have only one value ƒ (𝒙) for each 𝒙 in its domain, so no vertical line can intersect the graph of a function more than once. Function Not a function (Relation)  Domain & Range A function ƒ is like a machine that produces an output value ƒ(𝒙) in its range whenever we feed it an input value 𝒙 from its domain. The domain is the largest set of real 𝒙-values for which the formula gives real 𝒚-values. Function Graph  𝒇 𝟏 =𝟓 𝒇 𝟑 =𝟏 𝒇 𝟔 = 𝟏𝟎 𝒚=𝒇 𝒙 Domain: 𝟏, 𝟔 Range: 𝟏, 𝟏𝟎  Linear Function (Straight Line) Constant rate of change ≡ Slope Standard Equation 𝒚=𝒎𝒙+𝒌 General Equation 𝑨𝒙+𝑩𝒚+𝑪=𝟎 𝒎 = −𝑨 /𝑩 Find the equation of line with slope 4 and passes through (2,3). (2,3) 𝑦 =4𝑥+𝑘 ⇒ 3=8+𝑘 ⇒ 𝑘 = −5 ∴𝑦 =4𝑥−5 Another solution: 𝑦 − 𝑦1 𝑦−3 𝑥 − 𝑥1 =𝑚 ⇒ 𝑥−2 =4 ⇒ 𝑦−3=4𝑥−8 ∴𝑦 =4𝑥−5 Sketch the line 𝟐𝒚 + 𝟑 𝒙 − 𝟔 = 𝟎 and find its angle of inclination. 𝒚 Put 𝑦 = 0 ⇒ 𝑥-intercept = 2 Put 𝑥 = 0 ⇒ 𝑦-intercept = 3 −3 𝑚= = tan 𝜙 2 𝝓 𝒙 −3 ∴ 𝜙 = tan−1 ≅ 123.7° 2  Quadratic Function (Parabola) 𝒚 = 𝒙𝟐 Domain: −∞, ∞ ≡ ℝ Range: 0, ∞ Symmetry: Symmetric about 𝑦-axis Zeros: 𝑥 = 0 𝒚=𝟎 𝒙=? Zeros of a function ≡ 𝒙-intercepts  Bridges Projectiles Satellite Arch Dish  Absolute Value Function (Modulus) 𝒙 if 𝒙 ≥ 𝟎 𝒚= 𝒙 =ቊ −𝒙 if 𝒙 < 𝟎 𝒚 Domain: −∞, ∞ ≡ ℝ Range: 0, ∞ Symmetry: Symmetric about 𝑦-axis 𝒙 Zeros: 𝑥 = 0  Square Root Function 𝒚 𝒚= 𝒙 Domain: 0, ∞ 𝟗 ≠ ±𝟑 Range: 0, ∞ 𝒙 Zeros: 𝑥 = 0 𝒙𝟐 = 𝒙  Reciprocal Function 𝒚 𝟏 𝒚= 𝒙 Domain: ℝ − 0 Range: ℝ − 0 𝒙 Symmetry: Symmetric about origin Asymptotes: Vertical (𝑦-axis) or (𝑥 = 0) Horizontal (𝑥-axis) or (𝑦 = 0) Note that: 𝒚 𝟏 𝒚= 𝟐 𝒙 Domain: ℝ − 0 Range: 0, ∞ Symmetry: Symmetric about 𝑦-axis 𝒙 Asymptotes: Vertical (𝑦-axis) or (𝑥 = 0) Horizontal (𝑥-axis) or (𝑦 = 0)