Graphics Reference
In-Depth Information
x
x
1, sketch the graphs of f
39 If ABS( x )
and f ( x )
ABS and
ABS f .
40
The composite rule
If thefunction g : A R is continuous at x a and thefunction
f : g ( A )
R
is continuous at g ( a ), provethat f g is continuous at
x a .
41 Useqn 40 to provethat, if a real function f is continuous at x a ,
then the function ABS f is also continuous at a , where ABS is
defined as in qn 39.
42 For any real function f , wedefinea function f on thesame
domain as f by f
( x )
( f ( x )
f ( x )
). If f is continuous at a , prove
that f is also continuous at a .
Sketch the graph of f
x
when f ( x )
1.
43 For any real function f , wedefinea function f on thesame
domain as f by f
( x )
( f ( x )
f ( x )
). If f is continuous at a ,
provethat f is also continuous at a .
Sketch the graph of f
x
when f ( x )
1.
44 For any two real numbers a and b , wedefine
a
b
when b a ,
when a b .
max( a , b )
Provethat max( a , b )
( a b ) a b .
Provethat f ( x ) max( f ( x ), 0).
45 If real functions f and g havethesamedomain A , and areboth
continuous at x a A , provethat thefunction max( f , g ), defined
by max( f , g )( x ) max( f ( x ), g ( x )) is also continuous at x a .
46 If f ( x ) x and g ( x ) x , identify the function max( f , g ).
47 Let r : R 0 R denote the reciprocal function r ( x ) 1/ x , and let
f ( x ) 1 x . What is themaximum domain of definition of f r
and of r f ? Are both these functions continuous throughout their
domains? Use a computer or a graphic calculator to examine the
graphs of f , r f and f r .
48 Assuming that thesinefunction is continuous on R (see chapter 11),
determine the largest subset of R on which thefunction f of qn 19
is continuous.
49 Use a computer or a graphic calculator to examine the graph of
y x sin 1/ x .
 
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