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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

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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 Graphical Illustration E x * z * w * 5 * * f(w) * f(x) * f(z) * f(5) * 3 * 4 * - 9 D f f is a function

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More illustrations…. 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

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Find function values Example 1: Let f be the function with domain R such that f( x) = x 2 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 )

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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 Not Function

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Increasing, Decreasing and Constant Function TerminologyDefinitionGraphical Interpretation f is increasing over interval I f(x 1 ) < f(x 2 ) whenever x 1 < x 2 f is decreasing over interval I f(x 1 ) > f(x 2 ) whenever x 1 < x 2 f is constant over interval I f(x 1 ) = f(x 2 ) whenever x 1 = x 2 x1x1 x2x2 f(x 1 ) f(x 2 ) x y x1x1 x2x2 f(x 1 ) f(x 2 ) x y x1x1 x2x2 f(x 1 )f(x 2 ) x y

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

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Example 2: Sketch the graph that is decreasing on (,- 3] and [ 0, ), increasing on [ -3,0 ], f(-3) = 2 and f (2 ) = 0 Solution: -30 decreasing increasingdecreasing

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Interpretation of Domain and Range of a function f Domain 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 f

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Graphical Approach to Domain and Range Example 1: Find the natural domain and Range of the graph of the function f below The function f represents f (x ) = x 2. f is well defined everywhere in R. Therefore, Domain = R Range Domain Every value of f is non-negative ( greater than or equal to 0. Therefore, Range =

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

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

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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 = (4) f is well defined when and -5 Domain =

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Do all the Homework assigned in the syllabus for Section 3.4

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