VIEWS: 16 PAGES: 36 POSTED ON: 11/21/2011
2.1: Represent Relations and Functions Objectives: 1. To determine if a relation is a function 2. To find the domain and range of a relation or function 3. To classify and evaluate functions 4. To distinguish between discrete and continuous functions Vocabulary As a group, define Relation Function each of these without your book. Input Output Give an example of each word and Domain Range leave a bit of space Independent Dependent for additions and Variable Variable revisions. Relation A mathematical relation is the pairing up (mapping) of inputs and outputs. Relation A mathematical relation is the pairing up (mapping) of inputs and outputs. • Domain: the set of all input values • Range: the set of all output values Relation A mathematical relation is the pairing up (mapping) of inputs and outputs. What’s the domain and range of each relation? Exercise 1 Consider the relation given by the ordered pairs (3, 2), (-1, 0), (2, -1), (-2, 1), (0, 3). 1. Identify the domain and range 2. Represent the relation as a graph and as a mapping diagram Calvin and Hobbes! A toaster is an example of a function. You put in bread, the toaster performs a toasting function, and out pops toasted bread. Calvin and Hobbes! “What comes out of a toaster?” “It depends on what you put in.” – You can’t input bread and expect a waffle! What’s Your Function? A function is a relation in which each input Relations has exactly one output. • A function is a Functions dependent relation • Output depends on the input What’s Your Function? A function is a relation in which each input Relations has exactly one output. • Each output does not Functions necessarily have only one input How Many Girlfriends? If you think of the inputs as boys and the output as girls, then a function occurs when each boy has only one girlfriend. Otherwise the boy gets in BIG trouble. What’s a Function Look Like? What’s a Function Look Like? What’s a Function Look Like? What’s a Function Look Like? Exercise 2a Tell whether or not each table represents a function. Give the domain and range of each relationship. Exercise 2b The size of a set is called its cardinality. What must be true about the cardinalities of the domain and range of any function? Exercise 3 Which sets of ordered pairs represent functions? 1. {(1, 2), (2, 3), (3, 4), (3, 5), (5, 6)} 2. {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)} 3. {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1)} 4. {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)} Exercise 4 Which of the following graphs represent functions? What is an easy way to tell that each input has only one output? Vertical Line Test A relation is a function iff no vertical line intersects the graph of the relation at more than one point If it does, then an input has more than one output. Function Not a Function Domain and Range: Graphs • Domain: All x- values (L → R) Range: – {x: -∞ < x < ∞} Greater than or equal • Range: All y- to -4 values (D ↑ U) – {y: y ≥ -4} Domain: All real numbers Exercise 5 Determine the domain and range of each function. Domain and Range: Equations • Domain: What you are allowed to plug in for x. – Easier to ask what you can’t plug in for x. • Range: What you can get out for y using the domain. – Easier to ask what you can’t get for y. Exercise 6 Determine the domain of each function. 1. y = x2 + 2 2. y x 2 1 3. y x2 Protip: Domains of Equations When you have to find the domain of a function given its equation there’s really only two limiting factors: 1. The denominator of any fractions can’t be zero – Set denominators ≠ zero and solve 2. Square roots can’t be negative – Set square roots ≥ zero and solve Dependent Quantities Functions can also be thought of as dependent relationships. In a function, the value of the output depends on the value of the input. • Independent quantity: Input values, x- values, domain • Dependent quantity: Output value, which depends on the input value, y-values, range Exercise 7 The number of pretzels, p, that can be packaged in a box with a volume of V cubic units is given by the equation p = 45V + 10. In this relationship, which is the dependent variable? Function Notation In an equation, the dependent variable is usually represented as f (x). • Read “f of x” – f = name of function; x = independent variable – Takes place of y – f (x) does NOT mean multiplication! – f (3) means “the function evaluated at 3” where you plug 3 in for x. Exercise 8 Evaluate each function when x = −3. 1. f (x) = −2x3 + 5 2. g (x) = 12 – 8x Flavors of Functions Functions come in a variety of flavors. You will need to be able to distinguish a linear from a nonlinear function. Linear Function Nonlinear Function f (x) = 3x f (x) = x2 – 2x + 5 g (x) = ½ x – 5 g (x) = 1/x h (x) = 15 – 5x h (x) = |x| + 2 Analog vs. Digital • Analog: A signal created by some physical process – Sound, temperature, etc. – Contain an infinite amount of data • Digital: A numerical representation of an analog signal created by samples – Not continuous = set of points – Contain a finite amount of data Digital Signal Processing Original Analog Signal Digital Samples Digital Signal Processing Digital signal processing is about converting an analog signal into digital information, doing something to it, and usually converting it back into an analog signal. Continuous vs. Discrete • Continuous Function • Discrete Function – A function whose – A function whose graph consists of an graph consists of a set unbroken curve of discontinuous points Exercise 9 Determine whether each situation describes a continuous or a discrete function. 1. The cheerleaders are selling candy bars for $1 each to pay for new pom-poms. The function f (x) gives the amount of money collected after selling x bars. 2. Kenny determined that his shower head releases 1.9 gallons of water per minute. The function V(x) gives the volume of water released after x minutes. Assignment • P. 76-79: 1-3, 9-21 odd, 24, 34-38, 41, 45, 46, 51, 52 • P. 81: 1-9 odd • Working with Domain and Range Worksheet: 1-3, 6- 11, 13, 14, 15, 16, 18, 20, 21