Functions+Notes+_updated_.pdf

For example the domain of f x x is a 0 and the domain

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For example, the domain of f ( x ) = x is A = [0 , ) and the domain of g ( x ) = 2 - x is B = ( -∞ , 2], so the domain of ( f + g )( x ) = x + 2 - x is A B = [0 , 2]. The domain of fg is A B , but we can not divide by 0 and so the domain of f/g is { x A B | g ( x ) 6 = 0 } . In the example above, the domain of f/g is [0 , 2). Example 2.33. Let f ( x ) = x and g ( x ) = 1 - x . The table below collects the formulas of 3 f , f + g , f - g , fg , f/g and g/f , as well their domains.
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42 J. S´ anchez-Ortega There is another way of combining two functions to obtain a new function. It is called composition . Definition 2.34. Given two functions f and g , the composite function f g (also called the composition of f and g ) is defined by ( f g )( x ) = f ( g ( x )) In calculating ( f g )( x ) = f ( g ( x )) we first calculate g ( x ) and then calcu- late f of the result. We call g the inner function and f the outer func- tion of the composition. We can, of course, also calculate the composition ( g f )( x ) = g ( f ( x )), where f is the inner function, the one that gets cal- culated first, and g is the outer function, which gets calculated last. The functions f g and g f are usually quite different. The domain of f g is the set of all x in the domain of g such that g ( x ) is in the domain of f . In other words, ( f g )( x ) is defined whenever both g ( x ) and f ( g ( x )) are defined.
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2. Functions 43 Example 2.35. Let f ( x ) = x and g ( x ) = x + 1. The table below collects the formulas of the four composite functions f g , g f , f f , and g g , as well their domains. Note that, in general, f g 6 = g f . 2.12 Even and Odd Functions It often happens that the graph of a function will have certain kinds of sym- metry. The simplest kinds of symmetry relate the values of a function at x and - x . Definitions 2.36. Let f be a function. Suppose that - x belongs to the domain of f whenever x does. We say that f is an even function if f ( - x ) = f ( x ) for every x in the domain of f . We say that f is an odd function if f ( - x ) = - f ( x ) for every x in the domain of f . The names even and odd come from the fact that even powers such as x 0 = 1, x 2 , x 4 , . . . , x - 2 , x - 4 , . . . are even functions, and odd powers such as x 1 = x , x 3 , . . . , x - 1 , x - 3 , . . . are odd functions. Observe, for example, that ( - x ) 4 = x 4 and ( - x ) - 3 = - x - 3 . If f ( x ) is even (or odd), then so is any constant multiple of f ( x ) such as 2 f ( x ) or - 5 f ( x ) . Sums (and differences) of even functions are even; sums (and differences) of odd functions are odd . Examples 2.37. The function f ( x ) = 3 x 4 - 5 x 2 - 1 is even, since it is the sum of three even functions: 3 x 4 , - 5 x 2 , and - 1 = - x 0 . Similarly, 4 x 3 - (2 /x )
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44 J. S´ anchez-Ortega is an odd function. The function g ( x ) = x 2 - 2 x is the sum of an even function and an odd function and is itself neither even nor odd. The graph of an even function is symmetric about the y -axis . This means that if we have plotted the graph of f for x 0, we obtain the entire graph simply by reflecting this portion about the y -axis. The graph of an odd function is symmetric about the origin. If we already have the graph of f for x 0, we can obtain the entire graph by rotating this portion through 180 about the origin. Note that if an odd function f is defined at x = 0 , then f (0) = 0 .
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2. Functions
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