Exercises 1—14, to establish a big- O relationship, find witnesses C and k such that | f ( x ) | ≤ C | g ( x ) | whenever x > k . Show that ( x 2 + 1 ) / ( x + 1 ) is O ( x ) .
Exercises 1—14, to establish a big- O relationship, find witnesses C and k such that | f ( x ) | ≤ C | g ( x ) | whenever x > k . Show that ( x 2 + 1 ) / ( x + 1 ) is O ( x ) .
Use graphs to determine if each function f in Exercises 45–48
is continuous at the given point x = c.
[2 – x, if x rational
x², if x irrational,
45. f(x)
c = 2
x² – 3, if x rational
46. f(x) = { 3x +1, if x irrational,
c = 0
[2 – x, if x rational
47. f(x) = { x², if x irrational,
c = 1
x² – 3, if x rational
3x +1, if x irrational,
48. f(x) :
c = 4
If f(x)=-
– 3x, Show that f (x) satisfies the hypothesis of the Roll's Theorem on the interval [-3, 3]
3
Each of Exercises 25–36 gives a formula for a function y = f(x). In
each case, find f-x) and identify the domain and range of f-. As a
check, show that f(fx)) = f-"f(x)) = x.
25. f(x) = x
26. f(x) = x, x20
%3D
%3D
27. f(x) = x + 1
28. f(x) = (1/2)x – 7/2
30. f(x) = 1/r, x * 0
%3D
29. f(x) = 1/x, x>0
x + 3
31. f(x)
32. f(x) =
VE - 3
34. f(x) = (2x + 1)/5
2
33. f(x) = x - 2r, xs1
(Hint: Complete the square.)
* + b
x - 2'
35. f(x) =
b>-2 and constant
36. f(x) = x?
2bx, b> 0 and constant, xsb
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