Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 1 Introduction to Functions and Graphs Book cover will go here

2 Copyright © 2014, 2010, 2006 Pearson Education, Inc. Functions and Their Representations ♦ Learn function notation ♦ Represent a function four different ways ♦ Define a function formally ♦ Identify the domain and range of a function ♦ Use calculators to represent functions (optional) ♦ Identify functions ♦ Represent functions with diagrams and equations 1.3

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 3 Basic Concepts The following table lists the approximate distance y in miles between a person and a bolt of lightning when there is a time lapse of x seconds between seeing the lightning and hearing the thunder. The value of y can be found by dividing the corresponding value of x by 5. x (seconds) y (miles) 12345

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 4 Basic Concepts This table establishes a special type of relationship between two sets of numbers, where each valid input x in seconds determines exactly one output y in miles. The table represents or defines a function f, where function f computes the distance between an observer and a lightning bolt. x (seconds) y (miles) 12345

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 5 Basic Concepts The distance y depends on the time x, and so y is called the dependent variable and x is called the independent variable. The notation y = f(x) is used to emphasize that f is a function (not multiplication). It is read “y equals f of x” and denotes that function f with input x produces output y. That is, f (Input) = Output

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 6 Function Notation The notation y = f (x) is called function notation. The input is x, the output is y, and the name of the function is f. Output Name y = f (x) Input

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 7 Function Notation The variable y is called the dependent variable, and the variable x is called the independent variable. The expression f(20) = 4 is read “f of 20 equals 4” and indicates that f outputs 4 when the input is 20. A function computes exactly one output for each valid input. The letters f, g, and h are often used to denote names of functions.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 8 Domain and Range of a Function The set of all meaningful inputs x is called the DOMAIN of the function. The set of corresponding outputs y is called the RANGE of the function. A function f that computes the height after t seconds of a ball thrown into the air, has a domain that might include all the times while the ball is in flight, and the range would include all heights attained by the ball.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 9 Representation of Functions Functions can be represented by Verbal descriptions Tables Symbols Graphs

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 10 Verbal Representation (Words) In the lightning example, “Divide x seconds by 5 to obtain y miles.” OR “f calculates the number of miles from a lightning bolt when the delay between thunder and lightning is x seconds.”

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 11 Numerical Representation (Table of Values) Here is a table of the lightning example using different input-output pairs (the same relationship still exists): x (seconds) y (miles) Since it is inconvenient or impossible to list all possible inputs x, we refer to this type of table as a partial numerical representation.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 12 Symbolic Representation (Formula) In the lightning example, Similarly, if a function g computes the square of a number x, then

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 13 Graphical Representation (Graph) A graph visually pairs and x-input with a y-output. Using the lightning data:

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 14 Graphical Representation (Graph) The scatterplot suggests a line for the graph of f.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 15 Formal Definition of a Function A function is a relation in which each element of the domain corresponds to exactly one element in the range. The ordered pairs for a function can be either finite or infinite.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 16 Let a function f be represented symbolically by (a) Evaluate f(2), f(1), and f(a + 1) (b) Find the domain of f. Solution (a) Example: Evaluating a function and determining its domain

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 17 (b)The expression for f is not defined when the denominator x – 1 = 0, that is, when x = 1. So the domain of f is all real numbers except for 1. Example: Evaluating a function and determining its domain

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 18 Set-Builder Notation The expression {x | x ≠ 1} is written in set- builder notation and represents the set of all real numbers x such that x does not equal 1. Another example is {y | 1 < y < 5}, which represents the set of all real numbers y such that y is greater than 1 and less than 5.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 19 A function g is given by g(x) = x 2 – 2x, and its graph is shown. (a)Find the domain and range of g. (b)Use g(x) to evaluate g(–1). (c)Use the graph of g to evaluate g(–1). Example: Evaluating a function symbolically and graphically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 20 (a)The domain for g(x) = x 2 – 2x, is all real numbers. (b) g(–1) = (–1) 2 – 2(–1) = = 3 (c) Find x = –1 on the x-axis. Move upward to the graph of g. Move across (to the right) to the y-axis. Read the y-value: g(–1) = 3. Example: Evaluating a function symbolically and graphically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 21 A graph of (a) Evaluate f(1) (b) Find the domain and range of f. is shown. Example: Find the domain and range graphically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 22 (a) Start by finding 1 on the x-axis. Move up and down on the grid. Note that we do not intersect the graph of f. Thus f(1) is undefined. (b) Arrow indicates x and y increase without reaching a maximum. Domain is in green: D = {x | x ≥ 2} Range is in red: R = {y | y ≥ 0} Example: Find the domain and range graphically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 23 Identifying Functions: Vertical Line Test If every vertical line intersects a graph at no more than one point, then the graph represents a function. Note: If a vertical line intersects a graph more than once, then the graph does not represents a function.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 24 Use the vertical line test to determine if the graph represents a function. (b) Solution (a) (a) Yes(b) No Example: Identifying a function graphically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 25 Functions Represented by Diagrams and Equations There are two other ways that we can represent, or define, a function: Diagram Equation

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 26 Diagrammatic Representation (Diagram) Function Sometimes referred to as mapping; 1 is the image of 5; 5 is the preimage of 1.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 27 Diagrammatic Representation (Diagram) Not a function

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 28 Functions Defined by Equations The equation x + y = 1 defines the function f given by f(x) = 1 – x where y = f(x). Notice that for each input x, there is exactly one y output determined by y = 1 – x.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 29 Determine if y is a function of x. a) x = y 2 (b) y = x 2 – 2 Solution (a) If we let x = 4, then y could be either 2 or –2. So, y is not a function of x. The graph shows it fails the vertical line test. Example: Identifying a function

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 30 (b) y = x 2 – 2 Each x-value determines exactly one y-value, so y is a function of x. The graph shows it passes the vertical line test. Example: Identifying a function