Section 3.4 Objectives: Find function values Use the vertical line test Define increasing, decreasing and constant functions Interpret Domain and Range of a function Graphically and Algebraically

f D E f is a function Graphical Illustration Function: A function f is a correspondence from a set D to a set E that assigns to each element x of D exactly one value ( element ) y of E f Graphical Illustration x * z * w * 5 * * f(w) * f(x) * f(z) * f(5) * 3 * 4 * - 9 D E f is a function

More illustrations…. D E D E f is not a function Why? x * z * w * 5 * * f(w) * f(x) * f(z) * f(5) * 3 * 4 * - 9 D E f is not a function Why? x in D has two values x * z * w * 5 * * f(w) * f(x) * f(z) * f(5) * 3 * 4 * - 9 D E f is not a function Why? x in D has no values

Find function values Example 1: Let f be the function with domain R such that f( x) = x2 for every x in R. ( i ) Find f ( -6 ), f ( ), f( a + b ), and f(a) + f(b) where a and b are real numbers. Solution: Note: f ( a + b ) f( a ) + f ( b )

Vertical Line Test of functions Vertical Line test: The graph of a set of points in a coordinate plane is the graph of a function if every vertical line intersects the graph in at most one point Example: check if the following graphs represent a function or not Function Function Not Function Function

Increasing, Decreasing and Constant Function Terminology Definition Graphical Interpretation f is increasing over interval I f(x1) < f(x2) whenever x1 < x2 f is decreasing f(x1) > f(x2) f is constant f(x1) = f(x2) x1 = x2 x1 x2 f(x1) f(x2) x y x1 x2 f(x1) f(x2) x y x1 x2 f(x1) f(x2) x y

Example 1: Identify the interval(s) of the graph below where the function is Increasing Decreasing Solution: (a) Increasing (b) Decreasing:

Example 2: Sketch the graph that is decreasing on ( ,- 3] and [ 0, ), increasing on [ -3 ,0 ], f(-3) = 2 and f (2 ) = 0 Solution: decreasing increasing decreasing -3

Interpretation of Domain and Range of a function f is the Set of all x where f is well defined Range is the set of all values f( x ) Where x is in the domain

Graphical Approach to Domain and Range Example 1: Find the natural domain and Range of the graph of the function f below Range The function f represents f (x ) = x2. f is well defined everywhere in R. Therefore, Domain = R Every value of f is non-negative ( greater than or equal to 0. Therefore , Range = Domain

More illustrations of Domain and Range of a graph of a function f This graph does not end on both sides Domain = Range = These two graphs seem similar, but the domain and range are different This graph ends, it is also not defined at x = –2 and well defined at x =2 Domain = Range =

Class Exercise 1 Find the natural domain and range of the following graphs Domain = Range = Domain = Range = Domain = Domain = Range = Range =

Algebraic Approach to find the Domain of a function f Example 1: Find the natural domain of the following functions Solution: ( 1 ) f is a linear function. f is well-defined for all x. Therefore, Domain = R ( 2 ) f is a square root function. f is well defined when Domain = (3) f is well defined when Domain = and (4) f is well defined when Domain = -5

Do all the Homework assigned in the syllabus for Section 3.4