Outline. 1 The Role of Functions. 2 Polynomial Functions. 3 Power Functions. 4 Rational Functions. 5 Exponential & Logarithmic Functions

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1 Outline MS11: IT Mathematics Functions Catalogue of Essential Functions John Carroll School of Mathematical Sciences Dublin City University 1 The Role of Functions 3 Power Functions 4 Rational Functions 5 Eponential & Logarithmic Functions 6 The Role of Functions The Role of Functions Mathematical Models Outline Mathematical Model 1 The Role of Functions 3 Power Functions 4 Rational Functions 5 Eponential & Logarithmic Functions 6 Functions (/4) MS11: IT Mathematics John Carroll 3 / 58 A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life epectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. Functions (/4) MS11: IT Mathematics John Carroll 4 / 58

2 The Role of Functions Mathematical Models Mathematical Model Outline 1 The Role of Functions A mathematical model is never a completely accurate representation of a physical situation it is an idealization. A good model simplifies reality enough to permit mathematical calculations but is accurate enough to provide valuable conclusions. It is important to realize the limitations of the model. In the end, Mother Nature has the final say 3 Power Functions 4 Rational Functions 5 Eponential & Logarithmic Functions 6 Functions (/4) MS11: IT Mathematics John Carroll 5 / 58 Functions (/4) MS11: IT Mathematics John Carroll 6 / 58 Coordinate Geometry & Lines Coordinate Geometry & Lines (Cont d) Any point in the plane can be located by a unique ordered pair of numbers as follows: Draw lines through perpendicular to the - and y-aes. These lines intersect the aes in points with coordinates a and b as shown above. Functions (/4) MS11: IT Mathematics John Carroll 7 / 58 Then the point P is assigned the ordered pair (a, b). The first number a is called the -coordinate of P; the second number b is called the y-coordinate of P. We say that P is the point with coordinates (a, b), and we denote the point by the symbol P(a, b). Several points are labeled with their coordinates in the second plot above. Functions (/4) MS11: IT Mathematics John Carroll 8 / 58

3 Regions in the Plane Constant Functions: Slope m = 0 y y =.5 y = 1 y = 0 Formally {(, y) 0} {(, y) y = 1} {(, y) y < 1} Functions (/4) MS11: IT Mathematics John Carroll 9 / 58 y = 1. y = Functions (/4) MS11: IT Mathematics John Carroll 10 / 58 Linear Functions Linear Functions A function of the form y = f () = m + c, where m and c are constants, is called a linear function. y = 3 y y = y = y = When we say that y is a linear function of, we mean that the graph of the function is a line, so we can use the slope-intercept form of the equation of a line to write a formula for the function as y = 0 y = f () = m + c where m is the slope of the line and c is the y-intercept. A characteristic feature of linear functions is that they grow at a constant rate. Case 1: c = 0 Functions (/4) MS11: IT Mathematics John Carroll 11 / 58 Functions (/4) MS11: IT Mathematics John Carroll 1 / 58

4 Slope of a Straight Line Slope of a Straight Line (Cont d) The slope of a nonvertical line which passes through the points P 1 ( 1, y 1 ) and P (, y ) is m = y = y y 1 1 The slope of a vertical line is not defined. Functions (/4) MS11: IT Mathematics John Carroll 13 / 58 Functions (/4) MS11: IT Mathematics John Carroll 14 / 58 Straight Line: Equation & Inequality Slope as the Rate of Change Graph of equation 3 5y = 15 Graph of inequality + y > 5 Consider the graph of the linear function y = f () = 3 You will notice that whenever increases by 1, the value of y increases by 3. So y increases three times as fast as. Thus the slope of the graph, namely 3, can be interpreted as the rate of change of y with respect to. Functions (/4) MS11: IT Mathematics John Carroll 15 / 58 Functions (/4) MS11: IT Mathematics John Carroll 16 / 58

5 Polynomials Polynomials Polynomials Polynomial of Degree : Quadratic Function Definition A function is called a polynomial if P() = a n n + a n 1 n a + a 1 + a 0 where n is a nonnegative integer and the numbers a 0, a 1,..., a n are constants called the coefficients of the polynomial. The domain of any polynomial R = (, + ). If the leading coefficient a n 0, then the degree of the polynomial is n. For eample, the function is a polynomial of degree 6. P() = Functions (/4) MS11: IT Mathematics John Carroll 17 / 58 A polynomial of degree is of the form and is called a quadratic function. P() = a + b + c Its graph is always a parabola obtained by shifting the parabola y = a. The parabola opens upward if a > 0 and downward if a < 0. Functions (/4) MS11: IT Mathematics John Carroll 18 / 58 Quadratic Functions Quadratic Functions Quadratic functions Where is ( 3)( 5) = > 0? f () = a + b + c 3 The roots of this function are the solutions to the quadratic equation Questions to ask ( 3)( 5) = 0 Where is f () = ( 3)( 5) positive? Where is f () = ( a)( b), when a < b, positive? Where is g() = b = ( b)( + b), when 0 < b, positive? Where is h() = b = (b )(b + ) = g(), when 0 < b, positive? So i.e. = 3 and = 5. f () > 0 < 3 or > 5, f () < 0 3 < < 5. Functions (/4) MS11: IT Mathematics John Carroll 19 / 58 Functions (/4) MS11: IT Mathematics John Carroll 0 / 58

6 Quadratic Functions Quadratic Functions Quadratic functions Some Answers The quadratic function f () = ( a)( b), where a < b, is zero when = a and = b; positive if < a or > b; negative when a < < b. The quadratic function g() = b = ( b)( + b), where 0 < b, is zero when = ±b; positive if < b or > b; negative when b < < b. The quadratic function h() = b = (b )(b + ) = g(), where 0 < b, is zero when = ±b; positive if b < < b. negative when < b or > b; of Degree n A polynomial of degree 3 is of the form P() = a 3 + b + c + d, a 0 and is called a cubic function. Functions (/4) MS11: IT Mathematics John Carroll 1 / 58 Functions (/4) MS11: IT Mathematics John Carroll / 58 More General Polynomials More General Polynomials Polynomials: Recap Eamples of Polynomials A function f is a polynomial if f () = ( + 1)( 1) 1 f () = ( ) 4 ( + 1) 3 ( 1) 5 f () = a n n + a n 1 n a 1 + a o where n is a non negative integer, called the degree of the polynomial, and a 0, a 1,..., a n are real constants, called coefficients, with a n Functions (/4) MS11: IT Mathematics John Carroll 3 / 58 Functions (/4) MS11: IT Mathematics John Carroll 4 / 58

7 More General Polynomials More General Polynomials Polynomials Eamples of Polynomials Properties All polynomials have domain R = (, ). - y = m + b (with m 0) are polynomials of degree 1. Quadratic functions y = a + b + c (with a 0) are polynomials of degree. Cubic functions y = a 3 + b + c + d (with a 0) are polynomials of degree 3. If n is even, then f has even degree and if n is odd, then f has odd degree. A ball is thrown into the air with an initial velocity of 160ft/sec. It reaches a height, h, after t seconds: h(t) = 160t 16t ft. The supply and demand functions for a commodity are where P is the price per unit. P = Q S + 9Q S P = Q D 1Q D + 75 Functions (/4) MS11: IT Mathematics John Carroll 5 / 58 Functions (/4) MS11: IT Mathematics John Carroll 6 / 58 More General Polynomials More General Polynomials Evaluating Polynomials Graph of p() = Let p() = p is a polynomial of degree 3 (a cubic). We can, for eample, evaluate at = 0 to find p(0) = 7 (0) 3 1 (0) 8 (0) + 8 = 8 10 f () = While evaluating at = gives p() = 7 () 3 1 () 8 () + 8 = 0 In this case, = is called a root of the polynomial p. Note that p() = ( )(7 + 4). Functions (/4) MS11: IT Mathematics John Carroll 7 / 58 Functions (/4) MS11: IT Mathematics John Carroll 8 / 58

8 More General Polynomials Power Functions Roots of a Polynomial Outline In general, given any function f, we call a solution of the equation a root or zero of the function f. f () = 0 If = a is a root of a polynomial p, then p() = ( a)q(), for some polynomial q. We say that ( a) is a factor of p. A root = a is called a repeated root of the polynomial p, if p() = ( a) m s(), for some m and some polynomial s(). If p is a polynomial of degree n, then p has at most n real roots. 1 The Role of Functions 3 Power Functions 4 Rational Functions 5 Eponential & Logarithmic Functions 6 Functions (/4) MS11: IT Mathematics John Carroll 9 / 58 Functions (/4) MS11: IT Mathematics John Carroll 30 / 58 Power Functions Types of Power Functions Power Functions Types of Power Functions Power Functions Families of Power Functions A function of the form y = f () = n, where n is a constant, is called a power function. The graphs for n = 1,, 3, 4, 5 are displayed below: y y = y y = y y = 3 y y = 4 y y = 5 Functions (/4) MS11: IT Mathematics John Carroll 31 / 58 The general shape of the graph of f () = n depends on whether n is even or odd. Note that, as n increases, the graph of y = n becomes flatter near = 0 and steeper when 1. Functions (/4) MS11: IT Mathematics John Carroll 3 / 58

9 Power Functions Types of Power Functions Power Functions Types of Power Functions Power Functions: 1 n, n a positive integer Power Functions: Two Further Eamples 4 f () = 1 4 f () = Note that the domain of is [0, ) whose graph is the upper half of the parabola = y. The domain of 3 is R (every real number has a cube root). Functions (/4) MS11: IT Mathematics John Carroll 33 / 58 Functions (/4) MS11: IT Mathematics John Carroll 34 / 58 Rational Functions Rational Functions Definition Outline Rational Functions 1 The Role of Functions 3 Power Functions 4 Rational Functions 5 Eponential & Logarithmic Functions A rational function is a quotient or ratio of two polynomial functions: f () = p() q() where p() and q() are polynomials. The domain of a rational function is the set of all for which q() 0. 6 Functions (/4) MS11: IT Mathematics John Carroll 35 / 58 Functions (/4) MS11: IT Mathematics John Carroll 36 / 58

10 Rational Functions Eamples Rational Functions Eamples Eample of a Rational Function Rational Function with a Discontinuity The function f () = f () = 1 4 is a rational function with domain all real numbers. 4 Functions (/4) MS11: IT Mathematics John Carroll 37 / 58 Functions (/4) MS11: IT Mathematics John Carroll 38 / 58 Rational Functions Eamples Rational Functions Eamples Rational Function with a Discontinuity Rational Function with a Discontinuity The function 4 f () = is a rational function with domain all real numbers ecept when the denominator is zero: = 0 = 5 8. { So the domain is R\ (, 5 8 ) ( 58, ). 5 8 } = f () = Functions (/4) MS11: IT Mathematics John Carroll 39 / 58 Functions (/4) MS11: IT Mathematics John Carroll 40 / 58

11 Eponential & Logarithmic Functions Eponential & Logarithmic Functions Eponential Functions Outline Eponential Functions 1 The Role of Functions 3 Power Functions 4 Rational Functions 5 Eponential & Logarithmic Functions Functions of the form y = a, where a > 0 are called eponential functions, a is called the base. All eponential functions have domain R. If a 1, then the range of y = a is (0, ). 6 Functions (/4) MS11: IT Mathematics John Carroll 41 / 58 Functions (/4) MS11: IT Mathematics John Carroll 4 / 58 Eponential & Logarithmic Functions Eponential Functions Eponential & Logarithmic Functions Eponential Functions Eponential Function: f () = a Eponential Functions: Family Members In both cases, the domain is (, ), and the range is (0, ) Functions (/4) MS11: IT Mathematics John Carroll 43 / 58 Functions (/4) MS11: IT Mathematics John Carroll 44 / 58

12 Eponential & Logarithmic Functions Eponential Functions Eponential & Logarithmic Functions Eponential Functions The 3 kinds of Eponential Functions y = a Eponential Functions Application Suppose that e10, 000 euro is invested in an account with an interest rate of % per annum. The amount of money in the account after t years is P(t) = 10, 000(1.0) t. Functions (/4) MS11: IT Mathematics John Carroll 45 / 58 Functions (/4) MS11: IT Mathematics John Carroll 46 / 58 Eponential & Logarithmic Functions Logarithmic Functions Eponential & Logarithmic Functions Logarithmic Functions Logarithmic Functions Logarithmic Function: f () = log a Logarithmic functions are functions of the form y = log a where the base a 1 is a positive constant, are the inverse functions of the eponential functions. All logarithmic functions have domain (0, ) and range R. The domain is (0, ), and the range is (, ) Functions (/4) MS11: IT Mathematics John Carroll 47 / 58 Functions (/4) MS11: IT Mathematics John Carroll 48 / 58

13 Eponential & Logarithmic Functions Logarithmic Functions Eponential & Logarithmic Functions Logarithmic Functions Eponential & Logarithmic Functions a > 1 Logarithmic Functions Base a Functions (/4) MS11: IT Mathematics John Carroll 49 / 58 Functions (/4) MS11: IT Mathematics John Carroll 50 / 58 Eponential & Logarithmic Functions Logarithmic Functions Eponential & Logarithmic Functions Logarithmic Functions Natural Logarithmic Function y = ln Logarithmic Function y = ln( ) 1 Functions (/4) MS11: IT Mathematics John Carroll 51 / 58 Functions (/4) MS11: IT Mathematics John Carroll 5 / 58

14 Definition Outline 1 The Role of Functions Piecewise-defined functions are functions which are defined by different formulae in different parts of their domains. Simple Eample f () = 3 3 Power Functions 4 4 Rational Functions f () = f () = 5 Eponential & Logarithmic Functions Functions (/4) MS11: IT Mathematics John Carroll 53 / 58 4 Functions (/4) MS11: IT Mathematics John Carroll 54 / 58 Eamples Eamples Another eample of a function which is described by using different formulae on different parts of its domain is the absolute value function: {, if 0 f () = =, if < 0 Graphing The function, if < 0 f () =, if 0 1 1, if > 1. is defined on the real line but has values given by different formulas depending on the position of. y y = y = y = 1 0 y = 1 y = 1 Domain: (, ) Range: [0, ) Functions (/4) MS11: IT Mathematics John Carroll 55 / 58 Functions (/4) MS11: IT Mathematics John Carroll 56 / 58

15 Eamples Eamples The Greatest Integer function is the function whose value at is the greatest integer less than or equal to it is also called the integer floor function and is denoted. For eample 3. = 3 = 3 = 3.99, 3.7 = 4 = y y = Domain: (, ) Range: Z Functions (/4) MS11: IT Mathematics John Carroll 57 / 58 The Least Integer function is the function whose value at is the smallest integer greater than or equal to it is also called the integer ceiling function and is denoted. For eample 3. = 4 = 4 = , 3.7 = 3 = y y = Domain: (, ) Range: Z Functions (/4) MS11: IT Mathematics John Carroll 58 / 58

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