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MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS Functions in terms of Set Notation MATH0201 BASIC CALCULUS A Library of Basic Functions Polynomial Functions Functions Rational Functions Piecewise–Defined Function Dr. WONG Chi Wing Department of Mathematics, HKU Absolute Value Function Trigonometric Function Invertible Functions Reference § 2.1–4 of Barnett. MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS However, not every equation in two variables define a function. To summarize, a function can be represented by I Tables I Graphs I Formulas, or more general, an equation of two variables. Example 2 The equation x 2 + y 2 = 2 does not define a function. Given that x = 1, there corresponds the real values y = ±1. a Example 1 (Functions Defined by Equation) a The equation 3x + 2y = −1 defines a function. Every x is mapped to the number y = −(3x + 1)/2. a a Read Example 2(A) (p.47) Read Example 2(B) (p.47) Graph of x 2 + y 2 = 2 Theorem 3 (Vertical Line Test for Function (p.48)) An equation specifies a function if each vertical line in the coordinate system passes through, at most, one point on the graph of the equation; otherwise the equation does not specify a function. Graph of 3x + 2y = −1 1 / 42 2 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS Functions in terms of Set Notation Functions in terms of Set Notation Example 5 Let f : N → R be defined by f (x) = x 2 . Definition 4 (Function (p.46)) Domain : N Codomain : R Range : {1, 22 , 32 , 42 , . . .} = {n2 : n ∈ N} A function is a correspondence between two sets of elements such that to EACH element in the first set X , there corresponds ONE and ONLY ONE element in the second set Y . If we denote the function by f , then we may write f : X → Y . We call X and Y the domain and the codomain of the function respectively; and the set of the corresponding elements in Y is called the range of the function. 4 / 42 3 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS Functions in terms of Set Notation Functions in terms of Set Notation Example 6 If the graph of a function is given, then its range is the projection of the graph on the y –axis. Let f : R → R be defined by f (x) = x 2 . Domain : R Codomain : R Range : [0, ∞) Example 7 The range of function below is the blue segment. 5 / 42 6 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS Functions in terms of Set Notation Functions in terms of Set Notation If the domain of the function f (x) is not specified, then we shall assume the natural domain, i.e., the set of all real–number replacements of the independent variable that produce real values of the dependent variable. (p.48) Definition 9 (Increasing/Decreasing Functions (p.267)) We say that a function f is increasing on an interval (a, b) if f (x1 ) < f (x2 ) whenever a < x1 < x2 < b, Example 8 and f is decreasing on an interval (a, b) if Find the natural domain of 1 f (x1 ) > f (x2 ) whenever a < x1 < x2 < b, 1. f (x) = x 2 + x − 1, √ 2. g(x) = x − 1, Further, we say that f is monotone on an interval (a, b) if it is either increasing or decreasing there. 3. h(x) = 1/(x − 2). 1 Read Example 3 (p.48), 5 (p.50). 8 / 42 7 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS A Library of Basic Functions A Library of Basic Functions Polynomial Functions Polynomial Functions Example 10 (Cost and Demand Function (p.51)) Linear Function The cost C of manufacturing x units of a product may be given by A linear function takes the form f (x) = ax + b, C(x) = a + bx, where a, b > 0; while the price of a product p when x units of that product are available may be given by where a and b are fixed reals. p(x) = m − nx Domain : R Codomain : R Range :R Blank f (x) is increasing if a > 0 f (x) is decreasing if a < 0 where m, n > 0. Example 11 (Fahrenheit–Celsius Temperature Conversion) The Fahrenbeit temperature scale (y ◦ F ) is related to the Celsius temperature scale (x ◦ C) by 9 y = 32 + x. 5 9 / 42 10 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS A Library of Basic Functions A Library of Basic Functions Polynomial Functions Polynomial Functions Quadratic Function Example 12 (Velocity of a Free Falling Object) The velocity of a free falling object is given by A quadratic function takes the form v (t) = v (0) + gt f (x) = ax 2 + bx + c, where a, b, and c are fixed reals. where g is the acceleration due to gravity (in ms−2 ). Domain Codomain Range Example 13 (Zero Order Reaction) In a zero order reaction, the concentration of a reactant A is given by [A](t) = [A](0) − kt : : : R R [k , ∞) if a > 0 (−∞, k ] if a < 0 Blank Neither increasing nor decreasing where k is the reaction rate. 12 / 42 11 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS A Library of Basic Functions A Library of Basic Functions Polynomial Functions Polynomial Functions Example 14 (Revenue and Profit Function (p.51)) Example 15 (Distance Travel of a Free Falling Object) If the cost function and the demand function of a product are respectively given by The height of the Leaning Tower of Pisa is 55m. When Galilei dropped a cannon ball from the top, the height of the cannon ball after t seconds was C(x) = a + bx and p(x) = m − nx. h(t) = 55 − The revenue function is R(x) = xp(x) = mx − nx 2 . g 2 t . 2 Example 16 (Energy stored in a spring) The profit function is If a spring with natural length `0 is compressed/stretched to the length `, then the energy stored E is P(x) = R(x) − C(x) = −nx 2 + (m − b)x − a. E(`) = Read Example 7 (p.52). 13 / 42 k (` − `0 )2 . 2 14 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS A Library of Basic Functions A Library of Basic Functions Polynomial Functions Polynomial Functions Cube Function Third–degree Polynomial Function The cube function is defined by the formula A third–degree polynomial function takes the form f (x) = x 3 . f (x) = ax 3 +bx 2 +cx +d, Domain : R Codomain : R Range :R Blank f (x) = x 3 is increasing where a, b, c, and d are fixed reals. Domain : R Codomain : R Range :R They are neither increasing nor decreasing in general. 15 / 42 16 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS A Library of Basic Functions A Library of Basic Functions Polynomial Functions Polynomial Functions Square Root Function Example 17 (Compound Interest) The square root function is defined by the formula √ f (x) = x. If a principal P is invested at an annual rate r compounded annually, the the amount A at the end of the n–th year is P(r ) = A (1 + r )n . n=1: n=2: n=3: n=4: Domain : [0, ∞) Codomain : R Range : [0, ∞) Blank √ f (x) = x is increasing P(r ) = A(1 + r ) P(r ) = A 1 + 2r + r 2 P(r ) = A 1 + 3r + 3r 2 + r 3 P(r ) = A 1 + 4r + 6r 2 + 4r 3 + r 4 17 / 42 18 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS A Library of Basic Functions A Library of Basic Functions Polynomial Functions Rational Functions Cube Root Function Definition 18 (Rational Function (p.88)) A rational function is any function that can be written in the form The cube root function is defined by the formula √ f (x) = 3 x. f (x) = n(x) , d(x) d(x) 6= 0 where n(x) and d(x) are polynomials. Domain of f (x) is the set of all the reals x such that d(x) 6= 0. Domain : R Codomain : R Range :R Blank √ f (x) = 3 x is increasing 20 / 42 19 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS A Library of Basic Functions A Library of Basic Functions Rational Functions Piecewise–Defined Function Definition 20 (Piecewise–defined Function (p.64)) Example 19 (Lennard–Jones Potential) A piecewise–defined function is defined by rules for different parts of the domain. The potential energy V when two molecules approach each other from a long distance R apart is modeled by Example 21 (Example 6 (p.65)) Easton Utilities uses the rates shown in the table below to compute the monthly cost of natural gas for each customer. σ 12 − σ 6 R 6 . V (R) = 4ε · R 12 Note that I V (σ) = 0. I The minimum value of V is −ε, and it’s achieved when R = 21/6 σ. First 5 CCF : $ .7866 CCF-1 Next 35 CCF : $ .4601 CCF-1 Over 40 CCF : $ .2508 CCF-1 The cost of consuming x CCF is .7866x 3.933 + .4601(x − 5) C(x) = 20.0365 + .2508(x − 40) 21 / 42 if 0 ≤ x ≤ 5, if 5 < x ≤ 40, if x > 40. 22 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS A Library of Basic Functions A Library of Basic Functions Absolute Value Function Absolute Value Function Definition 22 The absolute value of a real number a, denoted by |a|, is defined by a if a ≥ 0, |a| = −a if a < 0. Theorem 24 For any real numbers a and b, 1. |a| ≥ 0 Example 23 2. |a| = | − a| 3. |ab| = |a||b| 1 4. a1 = |a| if a 6= 0 √ 5. a2 = |a|. (a) |3| = 3. (b) | − 2| = 2. (c) ||3 − 11| − |3|| = 5. 23 / 42 24 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS A Library of Basic Functions A Library of Basic Functions Absolute Value Function Absolute Value Function Example 25 Example 26 The graph of y = |2x − 1| is The graph of y = |x 2 − 3x + 2| is 25 / 42 26 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS A Library of Basic Functions A Library of Basic Functions Absolute Value Function Absolute Value Function Theorem 27 Theorem 30 For any a ≥ 0, solutions of |x| = a are a and −a. Let d > 0. Then the solutions of 1. |x| < d are exactly the real numbers satisfying −d < x < d, 2. |x| ≤ d are exactly the real numbers satisfying −d ≤ x ≤ d, 3. |x| > d are exactly the real numbers satisfying x > d or x < −d, 4. |x| ≥ d are exactly the real numbers satisfying x ≥ d or x ≤ −d. Example 28 The solution set of 1. |x| = 11 is {11, −11}. 2. |x| = 0 is {0}. 3. |x| = −1 is ∅. Example 29 Solve |3x − 2| = 1. 28 / 42 27 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS A Library of Basic Functions A Library of Basic Functions Absolute Value Function Trigonometric Function Angle An angle is formed by rotating a ray about its endpoint. Example 31 Solve the following inequalities (a) |2x − 1| < 3 (b) |3x + 2| ≥ 4. 29 / 42 I The endpoint of the ray is called the vertex. I The initial position of the ray is called the initial side. I The final position of the ray is called the terminal side. 30 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS A Library of Basic Functions A Library of Basic Functions Trigonometric Function Trigonometric Function Sine and Cosine of an Angle Definition 32 In the Cartesian plane, an angle is said to be in standard position if its vertex is at the origin and its initial side is along the positive x–axis. Consider the unit circle in a coordinate system centered at the origin. I Terminal side of an angle θ in standard position will pass through the circle at a point P. Angles can be represented by (real) numbers: I I A counterclockwise rotation has a positive measure while a clockwise rotation has a negative measure. If there is more than one complete rotation, then the magnitude of an angle can be greater than 360 degrees. I Abscissa of P is called the cosine of θ, denoted by cos θ. I Ordinate of P is called the sine of θ, denoted by sin θ. 32 / 42 31 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS A Library of Basic Functions Invertible Functions Trigonometric Function Definition 34 (One–to–one Function and Inverse Function (cf p.106–107)) Definition 33 (Sine, Cosine, and Tangent Functions) A function is said to be one–to–one if each range value corresponds to exactly one domain value. The function which sends every real number x to the sine of an angle with magnitude x ◦ is called the sine function, denoted by sin x. If f is a one–to–one function, then the inverse of f , denoted by f −1 , is the function formed by interchanging the independent and dependent variables for f . Thus, if (a, b) is a point on the graph of f , then (b, a) is a point on the graph of f −1 . Similarly, the function which sends every real number x to the cosine of an angle with magnitude x ◦ is called the cosine function, denoted by cos x. sin x The tangent function is defined by tan x = . cos x Remark 1 It’s clear that the domain of f is exactly the range of f −1 while the range of f is exactly the domain of f −1 . Theorem 35 If f is a monotone function on an interval (a, b), then it has inverse. 33 / 42 34 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS Invertible Functions Invertible Functions Finding Inverse Graphically Given that f has an inverse. Then the graph of f −1 is the reflection of the graph of f along the line y = x. Example 36 Both the linear function f (x) = ax + b (a 6= 0) and the cube function g(x) = x 3 have inverses. f −1 (x) = (x − b)/a g −1 (x) = √ 3 x 35 / 42 MATH0201 BASIC CALCULUS 36 / 42 MATH0201 BASIC CALCULUS Invertible Functions Invertible Functions Example 36 The square function f (x) = x 2 is not one–to–one and hence it has no inverse. Actually the reflection of its graph along y = x fails the vertical line test. Finding Inverse Analytically Let f be a one–to–one function. If a single x can be solved for the equation y = f (x) (in terms of y ), then f has an inverse. Example 37 Find the inverse of f (x) = √ 3x − 1, if any. Class Practice 7 Find the inverse of 4x 3 − 7, if any. 37 / 42 38 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS Invertible Functions Invertible Functions Example 38 While f (x) = x 2 is not one–to–one, its restriction to [0, ∞) is increasing and hence has an inverse. Example 39 The sine function f (x) = sin x has no inverse, but its restriction to [−90, 90] has an inverse.The inverse is denoted by arcsin x. 40 / 42 39 / 42 MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS Invertible Functions Invertible Functions Example 40 Example 41 The cosine function f (x) = cos x has no inverse, but its restriction to [0, 180] has an inverse.The inverse is denoted by arccos x. The tangent function f (x) = tan x has no inverse, but its restriction to (−90, 90) has an inverse.The inverse is denoted by arctan x. 41 / 42 42 / 42