Ch1 Mathematics Functions PDF

Dr. Ehab Magdy Lecturer of Applied Mathematics (Engineering Mechanics) Benha Faculty of Engineering Benha University Course Content ❑ Mathematics Functions ❑ Differentiation Rules ❑ Limits ❑ Integration Rules ❑ Probability  Chapter 1 Mathematics Functions...

Dr. Ehab Magdy Lecturer of Applied Mathematics (Engineering Mechanics) Benha Faculty of Engineering Benha University Course Content ❑ Mathematics Functions ❑ Differentiation Rules ❑ Limits ❑ Integration Rules ❑ Probability  Chapter 1 Mathematics Functions  Ch 1: Mathematics Functions What is a relation? is a connection between a set of inputs and a set of outputs An ordered pair consists of an x and y-coordinate (−3, −2), -3 (−1 , 0), (−1 , 4) -5 Input -1 Output (0 , 1), X -2 Y 0 0 (2 , −5), 2 1 (4 , 5), 4 4 (5 , 0) 5 5  Ch 1: Mathematics Functions Not every relation is a function. Every function is a relation. Let X and Y be two nonempty sets. A function from X into Y is a relation that associates with each element of X exactly one element of Y that we call the image of x. The Domain of a function is the set of all X The Range of a function is the set of all images Y  Ch 1: Mathematics Functions Focus on the X-coordinates, when given a relation If the set of ordered pairs has different X-coordinates, it IS A function If the set of ordered pairs has same X-coordinates, it is NOT a function Y-coordinates have no bearing in determining functions  Ch 1: Mathematics Functions {(–1, −7), (1, 0), (2, −3), (0, −8), (0, 5), (–2, −1)} Is this a function? Hint: Look only at the x-coordinates NO Is this still a relation? YES  Ch 1: Mathematics Functions {(0, −5), (1, −4), (2, −3), (3, −2), (4, −1), (5, 0)} Is this a relation? YES Is this a function? Hint: Look only at the x-coordinates YES  Ch 1: Mathematics Functions Is a relation a function? -3 -2 -3 √ One to One -1 -1 1 1 0 4 0 √ Many to One 𝐱 One to Many -2 -3 -2 𝐱 Many to Many -1 1 -1 1 4 0 4 Ch 1: Mathematics Functions  Ch 1: Mathematics Functions Linear Function ❑ Linear function (Line Equation) can be written in form: 𝒚 = 𝒎𝒙 + 𝒄 m and c are constant ❑ Example ❑Domain 𝒚=𝒙 𝒙 ∈ −∞, ∞ 𝒚 = 𝟎. 𝟓 𝒙 ❑Range 𝒚 = 𝟓𝒙 𝒚 ∈ −∞, ∞  Ch 1: Mathematics Functions Linear Function ❑ Example 𝒚 = 𝟕𝒙 + 𝟐 A A 𝑩 −𝟐 𝒙 0 𝟕 𝒚 2 0 B ❑Domain ❑Range 𝒙 ∈ −∞, ∞ 𝒚 ∈ −∞, ∞  Ch 1: Mathematics Functions Linear Function ❑ Example A 𝒚 = −𝒙 + 𝟑 A 𝑩 𝒙 0 𝟑 𝒚 3 0 ❑Domain ❑Range B 𝒙 ∈ −∞, ∞ 𝒚 ∈ −∞, ∞  Ch 1: Mathematics Functions Quadratic Function ❑ Quadratic function (Parabola Equation) can be written in form: 𝟐 𝒚= 𝒙+𝒉 +𝒌 k and h are constant ❑ Example 𝟐 𝒚=𝒙 ❑Domain ❑Range 𝒙 ∈ −∞, ∞ 𝒚 ∈ 𝟎 ,∞  Ch 1: Mathematics Functions Quadratic Function ❑ Quadratic function (Parabola Equation) can be written in form: 𝟐 𝒚= 𝒙+𝒉 +𝒌 k and h are constant ❑ Example 𝟐 𝒚 = (𝒙 + 𝟐) +𝟑 ❑Domain ❑Range -2 3 𝒙 ∈ −∞, ∞ 𝒚 ∈ 𝟑 ,∞  Ch 1: Mathematics Functions Quadratic Function ❑ Quadratic function (Parabola Equation) can be written in form: 𝟐 𝒚= 𝒙+𝒉 +𝒌 k and h are constant ❑ Example 𝟐 𝒚 = (𝒙 − 𝟒) −𝟓 ❑Domain ❑Range 4 -5 𝒙 ∈ −∞, ∞ 𝒚 ∈ −𝟓 , ∞  Ch 1: Mathematics Functions Quadratic Function ❑ Quadratic function (Parabola Equation) can be written in form: 𝟐 𝒚= 𝒙+𝒉 +𝒌 5 k and h are constant 4 ❑ Example 𝟐 𝒚 = −(𝒙 − 𝟓) + 𝟒 ❑Domain ❑Range 𝒙 ∈ −∞, ∞ 𝒚 ∈ −∞, 𝟒  Ch 1: Mathematics Functions Cubic Function ❑ Cubic function can be written in form: 𝟑 𝒚= 𝒙+𝒉 +𝒌 k and h are constant ❑ Example 𝟑 𝒚=𝒙 ❑Domain ❑Range 𝒙 ∈ −∞, ∞ 𝒚 ∈ −∞, ∞  Ch 1: Mathematics Functions Cubic Function ❑ Cubic function can be written in form: 𝟑 𝒚= 𝒙+𝒉 +𝒌 k and h are constant ❑ Example -2 3 𝟑 𝒚 = (𝒙 + 𝟐) +𝟑 ❑Domain ❑Range 𝒙 ∈ −∞, ∞ 𝒚 ∈ −∞, ∞  Ch 1: Mathematics Functions Cubic Function ❑ Cubic function can be written in form: 𝟑 𝒚= 𝒙+𝒉 +𝒌 k and h are constant ❑ Example 𝟑 5 𝒚 = −(𝒙 − 𝟓) + 𝟒 4 ❑Domain ❑Range 𝒙 ∈ −∞, ∞ 𝒚 ∈ −∞, ∞  Ch 1: Mathematics Functions Root Function ❑ Root function can be written in form: 𝒚= 𝒙+𝒉 +𝒌 k and h are constant ❑ Example 𝟏 𝒚= 𝒙= 𝒙𝟐 ❑Domain ❑Range 𝒙 ∈ 𝟎 ,∞ 𝒚 ∈ 𝟎 ,∞  Ch 1: Mathematics Functions Root Function ❑ Root function can be written in form: 𝒚= 𝒙+𝒉 +𝒌 k and h are constant ❑ Example 𝒚= 𝒙+𝟒 −𝟔 -6 ❑Domain ❑Range -4 𝒙 ∈ −𝟒 , ∞ 𝒚 ∈ −𝟔 , ∞  Ch 1: Mathematics Functions Root Function ❑ Root function can be written in form: 3 𝒚= 𝒙+𝒉 +𝒌 k and h are constant ❑ Example 𝒚=𝟓− 𝒙−𝟑 5 ❑Domain ❑Range 𝒙 ∈ 𝟑, ∞ 𝒚 ∈ −∞ , 𝟓  Ch 1: Mathematics Functions Root Function ❑ Root function can be written in form: 𝟑 𝒚= 𝒙+𝒉 +𝒌 3 k and h are constant ❑ Example 𝟑 5 𝒚= 𝒙−𝟑 +𝟓 ❑Domain ❑Range 𝒙 ∈ −∞, ∞ 𝒚 ∈ −∞, ∞  Ch 1: Mathematics Functions Exponential Function ❑ Exponential function can be written in form: 𝒎𝒙+𝒉 𝒚=𝒂 +𝒌 m ≠ 𝟎, 𝒂 , k , and h are constant ❑ Example 𝒙 𝟎.𝟓𝒙 𝟓𝒙 𝒚=𝟐 𝒚=𝟐 𝒚=𝟐 ❑Domain ❑Range 𝒙 ∈ −∞, ∞ 𝒚 ∈ 𝟎, ∞  Ch 1: Mathematics Functions Exponential Function ❑ Exponential function can be written in form: 𝒎𝒙+𝒉 𝒚=𝒂 +𝒌 m ≠ 𝟎, 𝒂 , k , and h are constant ❑ Example 𝒚=𝟐 𝒙 𝒚=𝟐 𝒙+𝟏 𝒚 = 𝟐𝒙−𝟏 ❑Domain ❑Range 𝒙 ∈ −∞, ∞ 𝒚 ∈ 𝟎, ∞  Ch 1: Mathematics Functions Exponential Function ❑ Exponential function can be written in form: 𝒎𝒙+𝒉 𝒚=𝒂 +𝒌 m ≠ 𝟎, 𝒂 , k , and h are constant ❑ Example ❑Range 𝒚 = 𝟐𝒙 𝒚 ∈ 𝟎, ∞ +1 𝒙 ∈ −∞, ∞ 𝒙 𝒚 = 𝟐 + 𝟏 𝒚 ∈ 𝟏, ∞ ❑ Domain 𝒙 -1 𝒚 = 𝟐 − 𝟏 𝒚 ∈ −𝟏, ∞  Ch 1: Mathematics Functions Exponential Function ❑ Exponential function can be written in form: 𝒎𝒙+𝒉 𝒚=𝒂 +𝒌 m ≠ 𝟎, 𝒂 , k , and h are constant ❑ Example 𝒚 = 𝟒−𝒙+𝟏 − 𝟑 ❑Domain ❑Range 𝒙 ∈ −∞, ∞ 𝒚 ∈ −𝟑, ∞  Ch 1: Mathematics Functions Natural Exponential Function ❑ Natural Exponential function can be written in form: 𝒎𝒙+𝒉 𝒚=𝒆 +𝒌 m ≠ 𝟎, 𝒂 , k , and h are constant 𝒚 = 𝒆−𝒙 𝒚 = 𝒆𝒙 𝒆 Euler’s Number ‘ ’ is a numerical constant used in mathematical calculations. The value of e is 2.7 𝟎 𝒆 =𝟏 𝒆∞ = ∞ 𝒆𝟏 = 𝒆 𝒆−∞ = 𝟎  Ch 1: Mathematics Functions Logarithmic Function ❑ Logarithmic functions are considered as the inverse of the exponential functions. ❑ Logarithmic functions can be written in form (The number 𝒂 is called the Base): 𝒚 = 𝒂𝒙 𝒙 = log 𝒂 𝒚 ❑ Logarithmic functions can be written in form (If 𝒂 = 𝟏𝟎): 𝒚 = 𝟏𝟎𝒙 𝒙 = log 𝟏𝟎 𝒚 = log 𝒚 ❑ Logarithmic functions can be written in form (If 𝒂 = 𝒆): 𝒙 𝒚=𝒆 𝒙 = log 𝒆 𝒚 = ln 𝒚  Ch 1: Mathematics Functions Logarithmic Function ❑ Logarithmic functions are considered as the inverse of the exponential functions. 𝒙 𝒚=𝒆 𝒚=𝒆 𝒙 ❑Domain ❑Range 𝒙 ∈ −∞, ∞ 𝒚 ∈ 𝟎, ∞ 𝒚 = ln 𝒙 ❑Domain ❑Range 𝒚 = ln 𝒙 𝒙 ∈ 𝟎, ∞ 𝒚 ∈ −∞, ∞  Ch 1: Mathematics Functions Trigonometric Function 𝒚 = sin 𝒙 𝒚 opposite ∆𝒚 1 𝐬𝐢𝐧 𝜽 = = 𝐜𝐬𝐜 𝜽 = hypotenuse ∆𝒓 𝐬𝐢𝐧 𝜽 ❑ Domain ❑ Range ∆𝒓 ∆𝒚 𝒙 ∈ −∞, ∞ 𝒚 ∈ −𝟏, 𝟏 𝜽 ∆𝒙 𝒙  Ch 1: Mathematics Functions Trigonometric Function 𝒚 = cos 𝒙 𝒚 The cosine of an angle: is defined as the sine of the complementary angle adjacent ∆𝒙 1 𝐜𝐨𝐬 𝜽 = hypotenuse = ∆𝒓 𝐬𝐞𝐜 𝜽 = 𝐜𝐨𝐬 𝜽 ∆𝒓 ∆𝒚 ❑ Domain ❑ Range 𝜽 𝒙 ∈ −∞, ∞ 𝒚 ∈ −𝟏, 𝟏 ∆𝒙 𝒙  Ch 1: Mathematics Functions Trigonometric Function 𝒚 = tan 𝒙 𝒚 opposite ∆𝒚 1 𝒕𝒂𝒏 𝜽 = = 𝒄𝒐𝒕 𝜽 = adjacent ∆𝒙 𝒕𝒂𝒏 𝜽 ❑ Domain ❑ Range ∆𝒓 𝒙 ∈ −∞, ∞ 𝒚 ∈ −∞, ∞ ∆𝒚 𝜽 ∆𝒙 𝒙  Next to Ch 2 Differentiation Rules