set2 - Lecture 4 Some basic properties of functions...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 y-axis (or inversion w.r.t. the y-axis). 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...
View Full Document

Page1 / 15

set2 - Lecture 4 Some basic properties of functions...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online