P.3 Functions and Their Graphs KNOW everything in P.3 !! assumed knowledge of Trigonometric functions (refer to section D.3 of Appendix D - student CD-ROM) KNOW graphs (8) in Figure P.27 (text p22) function notation Definition of a function Domain and range of a function Sketch f (x) Transformations of f (x) Classify ( * algebraic or transcendental ) functions and * even, odd or neither recognize combinations of functions helpful with definite integrals in Ch. 4 ! topics that I will go over in class! 1 Functions and Function Notation Defs. A relation is a set of ordered pairs. The set of first entries of the ordered pairs is the domain (D) of the relation. The set of second entries is the range (R) of the relation. A function is a relation that assigns to each number x in D exactly one number y in R. real lif e! the area A of a circle is a function of its radius r the cost C of a cell phone bill is a function of the number of texting minutes used m the population P of wolves is a function of the food supply s Terminology x (r, m, s, . ) is the independent variable (argument) y (A, C, P, . ) is the dependent variable (image) 2 Function (Euler's) Notation The number f (x), read "f of x" is called the value of f at x, or the image of x under f. Instead of y = x2 , 2 we write f (x) = x . function independent name variable FUNCTION NOTATION p. 19 The word function was first used by Gottfried Wilhelm Leibniz in 1694 . Forty years later, Leonhard Euler used the word function to . He introduced the notation y = f (x). The Graph of a Function y (x, f (x)) ) f (x y = f (x) ­ directed distance from x­axis x x ­ directed distance from y­axis 3 Evaluating a Function Examples: 1. for f (x) = x + 3 , evaluate a.) f(­2) b.) f(c + 1) 2 f(x + x) ­ f(x) , x for f (x) = x + 7 , evaluate = 0 2. x difference quotient! 4 Visually recognizing a function: if every vertical line in the xy­plane intersects the vertical line test graph of a relation in at most one point, then the relation is a function of x. [see Figure P.26 p22] KNOW graphs (8) in Figure P.27 (text p22) Identity function Squaring function Cubing function Square root function Absolute value function Rational function Sine function Cosine function A one­to­one function: ­1 [a function f whose inverse f (x) is also a function!] vertical AND horizontal line tests if every vertical AND horizontal line in the xy­plane intersects the graph of a function in at most one point , then the function is a one­to­one function. 5 The Domain (D) and Range (R) of a Function to define the domain: (1.) never divide by zero! (2.) deal with real­valued functions exclusively Examples: Find the domain and range. 2 1. f (x) = x ­ 5 1 2. h(t) = 2 ­ t piecewise­ x + 4 , x = 5 defined 3. function! f (x) = 2 { (x ­ 5) , x > 5 and evaluate f (­3) and f (10). 6 Assignment Read p19­26: YOU are responsible for all material! Do p27­28: #1­11 odd, 15­23 odd, #39­45 all 7 Transformations of Functions E X P L O R A T I O N p. 23 (a) y = | x + 2 | (b) y = sin(x) +1 (c) y = (x ­ 2)2 ­ 1 (d) y = (x + 1)3 + 2 Basic Types of Transformations (c > 0) p. 23 + vertical: . stretch y = c f (x), c > 1 . shrink y = c f (x), 0 < c < 1 horizontal: . stretch y = f (c x), 0 < c < 1 . shrink y = f (c x), c > 1 8 Classifications of Functions many real­world phenomena mathematical models Elementary Functions Algebraic (polynomial, radical, and rational ) p (x) q (x) f (x) = , = 0/ q (x) Trigonometric (sine, cosine, tangent, . .) ­ Appendix D transcendental { Exponential and Logarithmic Students study/review (p. 24) : ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ degree constant term coefficients leading coefficient test 9 Combinations of Functions Two functions can be combined in various ways to create new functions! sum: ( f + g)(x) = f (x) + g(x) difference: ( f ­ g)(x) = f (x) ­ g(x) . product: ( f g)(x) = f (x) g(x) f (x) : ( f g)(x) = quotient / g(x) composition ­ a function of a function f ( g(x)) read " f of g of x " f g read " f circle g " ( ( f g (x) = f ( g(x)) note: order of composition is important! ( ( f g (( x) = / (g f (x) 10 2 Examples: Given f (x) = x and g (x) = x ­ 2 , find : a.) g f b.) f g c.) g ( f (­3)) 11 Test for Even and Odd Functions The function y = f(x) is: even if f (­x) = f (x) odd if f (­x) = ­ f (x) Examples: Determine whether each function is even, odd, or neither. 2 1. f (x) = x 2 2. f (x) = x + 1 3 3. f (x) = x 3 4. f (x) = x + 1 3 5. f (x) = x + sin x recall: sin(­ ) = ­ sin( ) 12 Visualizing Even and Odd Functions E X P L O R A T I O N Describe a way to identify a function as even or odd by inspecting the graph! 2 3 f (x) = x f(x) = x 3 2 3 f (x) = x + sin x f (x) = x + 1 f (x) = x + 1 Even symmetry to y­axis Odd symmetry to origin 13 Assignment Do p28­29: #47­57 all, 59­62 all, #71­76 all 14.