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Unformatted text preview: Lecture 4 Some basic properties of functions “Vertical line test” From the definition of function, for each x ∈ D ( f ), there is exactly one value f ( x ). For functions f : R → R , this implies that a vertical line passing through a point x ∈ D ( f ) will intersect the graph of f at only one point. The situation at the left is acceptable, whereas the situation at the right is not. x y x y For this reason, the set of points ( x,y ) ∈ R 2 that satisfy the relation y = x 2 correspond to the graph of the function f ( x ), whereas the set of points ( x,y ) that satisfy x = y 2 do not correspond to a function g ( x ). (Note that we consider x to be the “official” input variable here.) x y x y y = x 2 x = y 2 As you may well know, the way to get around this “impasse” in the second case is to define two functions from this set of points: the functions g 1 ( x ) = √ x and g 2 ( x ) = − √ x , where it is understood that the “ √ ” operator yields the positive square root. 26 x y y = g 1 ( x ) = √ x y = g 2 ( x ) = √ x Symmetry If a function f satisfies the following relation, f ( − x ) = f ( x ) , for all x ∈ D ( f ) , (1) then f is called an even function . Examples: 1. f ( x ) = x 2 , f ( x ) = x 4 ; in fact, f ( x ) = x 2 n , n ∈ Z = {··· , − 2 , − 1 , , 1 , 2 , ···} , 2. f ( x ) = cos x , 3. f ( x ) = x 2 + 12 x 6 . 4. f ( x ) = 2 x 20 + 15cos x . A consequence of this is that the graph of f is symmetric with respect to the yaxis (or inversion w.r.t. the yaxis). y x a − a f ( a ) y = f ( x ) Graph of an even function f ( x ) 27 If a function f satisfies the following relation, f ( − x ) = − f ( x ) , for all x ∈ D ( f ) , (2) then f is called an odd function . Examples: 1. f ( x ) = x , f ( x ) = x 3 ; in fact, f ( x ) = x 2 n − 1 , n ∈ Z = {··· , − 2 , − 1 , , 1 , 2 , ···} , 2. f ( x ) = sin x , 3. f ( x ) = x + 12 x 5 . 4. f ( x ) = 2 x 19 + 15sin x . A consequence of this is that the graph of f is symmetric with respect to the origin (0 , 0) (or inversion w.r.t. the origin). Another way to state this property is that the graph of f is unchanged if we rotate it about the origin by 180 ◦ . y x a a f ( a ) f ( a ) y = f ( x ) Graph of an odd function f ( x ) You’ll note that the above graph, which is supposed to represent an odd function in general, passes through the origin (0 , 0). This is not a lack of generality but a necessary property of odd functions – assuming that they are defined at x = 0 – as we now show. Suppose that f is an odd function and f (0) is defined. Then, from the definition of an odd function, Eq. (2), setting x = 0, we have f ( − 0) = − f (0) . (3) 28 But the LHS is simply f (0). This means that f (0) = − f (0) , (4) which implies that 2 f (0) = 0 . (5) Therefore f (0) = 0. Note that this is true only for odd functions that are defined at x = 0. We may not make this conclusion if f (0) is undefined. An example is the odd function, f ( x...
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 Fall '08
 SPEZIALE
 Calculus, Inverse function

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