10 Quadratic Equations

10.5 Introduction to Functions Objectives 1. Understand the definition of a relation. 2. Understand the definition of a function. 3. Decide whether an equation defines a function. 4. Use function notation. 5. Apply the function concept in an application.

Understand the Definition of a Relation In an ordered pair (x, y), x and y are called the components of the ordered pair. Any set of ordered pairs is called a relation. The set of all first components in the ordered pairs of a relation is the domain of the relation, and the set of all second components in the ordered pairs is the range of the relation.

Understand the Definition of a Relation Example 1 Identify the domain and the range for the relation {(–2,5), (1,6), (1,3), (3,3), (5,3)}. The domain is the set of all first components, and the range is the set of all second components. The domain is {–2, 1, 3, 5} and the range is {3, 5, 6}. The relation in Example 1 is not a function, because the first component, 1, corresponds to more than one second component.

Understand the Definition of a Function A function is a set of ordered pairs in which each distinct first component corresponds to exactly one second component.

Understand the Definition of a Function Example 2 Determine whether each relation is a function. (a) {(1,2), (2,2), (3,4), (4,3), (5,6), (6,6)} This relation IS a function because each number in the domain corresponds to only one number in the range. (b) {(1,2), (2,2), (3,4), (4,3), (6,5), (6,6)} This relation IS NOT a function because 6 in the domain corresponds to both 5 and 6 in the range.

Decide Whether an Equation Defines a Function Vertical Line Test If a vertical line intersects a graph in more than one point, the graph is not the graph of a function.

Decide Whether an Equation Defines a Function Example 3 Decide whether each relation graphed or defined is a function. (a) y = 3x – 2 Use the vertical line test. Any vertical line intersects the graph just once, so this is the graph of a function.

Decide Whether an Equation Defines a Function Example 3 (concluded) Decide whether each relation graphed or defined is a function. (b) x2 + y2 = 4 The vertical line test shows that this is not the graph of a function; a vertical line could intersect the graph twice.

Use Function Notation The letters f, g, and h are commonly used to name functions. For example, the function with defining equation y = 3x +5 may be written f (x) = 3x + 5, where f (x) is read “ f of x.” The notation f (x) is another way of writing y in a function. For the function defined by f (x) = 3x + 5, if x = 7 then f (7) = 3·7 + 5 = 26. Read this result, f (7) = 26, as “ f of 7 equals 26.” The notation f (7) means the value of y when x is 7. The statement f (7) = 26 says that the value of y is 26 when x is 7. It also indicates the point (7, 26) lies on the graph of f.

Use Function Notation CAUTION The notation f (x) does not mean f times x. The symbol f (x) means the value of the function f when evaluated for x. It represents the y-value that corresponds to x. Function Notation In the notation f (x), f is the name of the function, x is the domain value, and f (x) is the range value y for the domain value x.

Using Function Notation Example 4 For the function defined by f (x) = –2x2 + 3x – 1, find the following. (a) f (5) (b) f (0) Substitute 5 for x. Substitute 0 for x. f (5) = –2(5)2 + 3(5) – 1 f (0) = –2(0)2 + 3(0) – 1 f (5) = –50 + 15 – 1 f (0) = – 1 f (5) = –36

Apply the Function Concept in an Application Example 5 The profits for Jeannie’s Jeans is given in the table below. Year 2001 2002 2003 2004 2005 2006 Profit (in millions) $0.75 $1.6 $2.1 $3.8 $4.7 (a) If we choose years as the domain elements and profits as the range elements, does this relation represent a function? Why/why not? Yes, this is a function because each year corresponds to exactly one profit.

Apply the Function Concept in an Application Example 5 (concluded) The profits for Jeannie’s Jeans is given in the table below. Year 2001 2002 2003 2004 2005 2006 Profit (in millions) $0.75 $1.6 $2.1 $3.8 $4.7 (b) Find f (2002) and f (2005). (c) For what x value does f (x) equal $2.1 million? f (2002) = $1.6 million f (2003) = $2.1 million f (2005) = $3.8 million