Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall
Chapter 3 Graphs and Functions
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 3.2 Introduction to Functions
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Relation, Domain, and Range A relation is a set of ordered pairs. The domain of the relation is the set of all first components of the ordered pairs. The range of the relation is the set of all second components of the ordered pairs.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Determine the domain and range of the relation {(4,9), (–4,9), (2,3), (10, –5)} Solution The domain is the set of all the first coordinates of the ordered pairs. Domain: 4, –4, 2, 10}. The range is the set of all second coordinates of the ordered pairs. Range: 9, 3, –5}. Example 1a
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Input (Animal) Polar Bear Cow Chimpanzee Giraffe Gorilla Kangaroo Red Fox Output (Life Span) Find the domain and range of the following relation. Example 1b Domain: {Polar Bear, Cow, Chimpanzee, Giraffe, Gorilla, Kangaroo, Red Fox}. Range: {20, 15, 10, 7}.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Some relations are also functions. A function is a relation in which each first component in the ordered corresponds to exactly one second component.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Given the relation {(4,9), (–4,9), (2,3), (10, –5)}, is it a function? Solution Since each element of the domain is paired with only one element of the range, it is a function. Note: It is okay for a y-value to be assigned to more than one x-value, but an x-value cannot be assigned to more than one y-value (it has to be assigned to ONLY one y-value). Example 2
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Is the relation y = x 2 – 2x a function? Solution Each element of the domain (the x-values) would produce only one element of the range (the y- values), it is a function. Example 3
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Is the relation x 2 – y 2 = 9 a function? Solution Each element of the domain (the x-values) would correspond with 2 different values of the range (both a positive and negative y-value), the relation is NOT a function. Example 4
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Vertical Line Test If no vertical line can be drawn so that it intersects a graph more than once, the graph is the graph of a function.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use the vertical line test to determine whether the graph to the right is the graph of a function. x y Yes, this is the graph of a function since vertical line will intersect this graph more than once. Example 5
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use the vertical line test to determine whether the graph to the right is the graph of a function. x y Example 5 Yes, this is the graph of a function since vertical line will intersect this graph more than once.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use the vertical line test to determine whether the graph to the right is the graph of a function. No, this is not the graph of a function. Vertical lines can be drawn that intersect the graph in two points. x y Example 5
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Since the graph of a linear equation is a line, all linear equations are functions, except those whose graph is a vertical line Note: An equation of the form y = c is a horizontal line and IS a function. An equation of the form x = c is a vertical line and IS NOT a function.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Find the domain and range of the function graphed to the right. x y Domain: 3 ≤ x ≤ 4 Domain Range: 4 ≤ y ≤ 2 Range Example 6
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Find the domain and range of the function graphed to the right. x y Domain: all real numbers Domain Range: y ≥ – 2 Range Example 6
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Function Notation To denote that y is a function of x, we can write y = f(x) (Read “f of x”) Function notation This notation means that y is a function of x or that y depends on x. For this reason, y is called the dependent variable and x the independent variable.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall If f(x) = x 2 – 2x, find f(–3). Solution f(–3) = (–3) 2 – 2(–3) = 9 – (–6) = 15 Example 7 f(x) = x 2 – 2x
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Given the graph of the following function, find each function value by inspecting the graph. f(5) = 6 x y f(x)f(x) f(4) = 3 f( 5) = 11 f( 6) = –6 Example 8