MAT 125 – Applied Calculus 2.1 Functions and Their Graphs

Today’s Class  We will be learning the following concepts:  Functions  Determining the Domain of a Function  Graphs of Functions  The Vertical Line Test Dr. Erickson 2.1 Functions and Their Graphs 2

Functions  Function: A function is a rule that assigns to each element in a set A one and only one element in a set B.  The set A is called the domain of the function.  It is customary to denote a function by a letter of the alphabet, such as the letter f. Dr. Erickson 2.1 Functions and Their Graphs 3

Functions  The element in B that f associates with x is written f(x) and is called the value of f at x.  The set of all the possible values of f(x) resulting from all the possible values of x in its domain, is called the range of f(x).  The output f(x) associated with an input x is unique: Each x must correspond to one and only one value of f(x). Dr. Erickson 2.1 Functions and Their Graphs 4

Example 1  Let the function f be defined by the rule. Dr. Erickson 2.1 Functions and Their Graphs 5

Determining the Domain of a Function  Suppose we are given the function y = f (x). Then, the variable x is called the independent variable. The variable y, whose value depends on x, is called the dependent variable.  To determine the domain of a function, we need to find what restrictions, if any, are to be placed on the independent variable x.  In many practical problems, the domain of a function is dictated by the nature of the problem. Dr. Erickson 2.1 Functions and Their Graphs 6

Example 2  Find the domain of the function. Dr. Erickson 2.1 Functions and Their Graphs 7

Graphs of Functions  If f is a function with domain A, then corresponding to each real number x in A, there is precisely one real number f (x).  Thus, a function f with domain A can also be defined as the set of all ordered pairs (x, f (x)) where x belongs to A.  The graph of a function f is the set of all points (x, y) in the xy-plane such that x is the domain of f and y = f (x). Dr. Erickson 2.1 Functions and Their Graphs 8

Example 3  The graph of a function f is shown below: Domain Range Dr. Erickson 2.1 Functions and Their Graphs 9 a.Find f (0). b.Find the value of x for which f (x) = 3. c.Find the domain of f. d.Find the range of f.

Example 4  Sketch the graph of the function with the given rule. Find the domain and range of the function. Dr. Erickson 2.1 Functions and Their Graphs 10

Vertical Line Test  A curve in the xy-plane is the graph of a function y = f (x) if and only if each vertical line intersects it in at most one point.  a. Function  Not a function Dr. Erickson 2.1 Functions and Their Graphs 11

Example 5  Use the vertical line test to determine whether the graph represents y as a function of x. Dr. Erickson 2.1 Functions and Their Graphs 12

Example 6 Dr. Erickson 2.1 Functions and Their Graphs 13

Example 7 Dr. Erickson 2.1 Functions and Their Graphs 14

Example 8 Dr. Erickson 2.1 Functions and Their Graphs 15

Next Class  We will discuss the following concepts:  The Sum, Difference, Product, and Quotient of Functions  Composition of Functions  Mathematical Models  Rational and Power Functions  Economic Models  Please read through Section 2.2 – The Algebra of Functions and Section 2.3 – Function and Mathematical Models in your text book before next class. Dr. Erickson 2.1 Functions and Their Graphs 16