# (a) Prove that g x 5 decreasing in this interval in .NET Printing QR Code JIS X 0510 in .NET (a) Prove that g x 5 decreasing in this interval

3.5. (a) Prove that g x 5 decreasing in this interval
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p 9 x is strictly decreasing in 0 @ x @ 9. (c) Does g x have a single-valued inverse
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(b) Is it monotonic
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(a) p strictly decreasing 1 > g x g x is If x1 < x2 then 9 x1 > 9 x2 , p p if g x p 2 whenever x1 < x2 . 9 x1 > 9 x2 , 5 9 x1 > 5 9 x2 showing that g x is strictly decreasing. (b) Yes, any strictly decreasing function is also monotonic decreasing, since if g x1 > g x2 it is also true that g x1 A g x2 . However, if g x is monotonic decreasing, it is not necessarily strictly decreasing. p p (c) If y 5 9 x, then y 5 9 x or squaring, x 16 10y y2 y 2 8 y and x is a single-valued function of y, i.e., the inverse function is single-valued. In general, any strictly decreasing (or increasing) function has a single-valued inverse (see Theorem 6, Page 47). The results of this problem can be interpreted graphically using the gure of Problem 3.4.
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& 3.6. Construct graphs for the functions integer @ x. (a) f x
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x sin 1=x; 0;
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x>0 , x 0
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(b) f x x greatest
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(a) The required graph is shown in Fig. 3-8. Since jx sin 1=xj @ jxj, the graph is included between y x and y x. Note that f x 0 when sin 1=x 0 or 1=x ; m, m 1; 2; 3; 4; . . . ; i.e., where x 1=; 1=2; 1=3; . . . . The curve oscillates in nitely often between x 1= and x 0. p (b) The required graph is shown in Fig. 3-9. If 1 @ x < 2, then x 1. Thus 1:8 1, 2 1, 1:99999 1. However, 2 2. Similarly for 2 @ x < 3, x 2, etc. Thus there are jumps at the integers. The function is sometimes called the staircase function or step function.
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3.7. (a) Construct the graph of f x tan x. (b) Construct the graph of some of the in nite number of branches available for a de nition of tan 1 x. (c) Show graphically why the relationship of x
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CHAP. 3]
FUNCTIONS, LIMITS, AND CONTINUITY
f (x)
f (x)
1/2p
_3 _2 _1
1 2 3 4 5
Fig. 3-8
Fig. 3-9
to y is multivalued. (d) Indicate possible principal values for tan 1 x. (e) Using your choice, evaluate tan 1 1 .
(a) The graph of f x tan x appears in Fig. 3-10 below.
y = f (x) = tan x
(x) = tan 1x
3p/2 p
p/2 _ p _ p/2 p/2 p 3p/2 2p
_ p/2 _p
Fig. 3-10
Fig. 3-11
(b) The required graph is obtained by interchanging the x and y axes in the graph of (a). The result, with axes oriented as usual, appears in Fig. 3-11 above. (c) In Fig. 3-11 of (b), any vertical line meets the graph in in nitely many points. Thus, the relation of y to x is multivalued and in nitely many branches are available for the purpose of de ning tan 1 x.
(d) To de ne tan 1 x as a single-valued function, it is clear from the graph that we can only do so by restricting its value to any of the following: =2 < tan 1 x < =2; =2 < tan 1 x < 3=2, etc. We shall agree to take the rst as de ning the principal value. Note that no matter which branch is used to de ne tan 1 x, the resulting function is strictly increasing. (e) tan 1 1 =4 is the only value lying between =2 and =2, i.e., it is the principal value according to our choice in d .
3.8. Show that f x
If y
p p x 1 then x 1 y 1 x or squaring, x 1 2 y2 2 x 1 y 1 x 0, a polynomial x 1 equation in y whose coe cients are polynomials in x. Thus f x is an algebraic function. However, it is not the quotient of two polynomials, so that it is an irrational algebraic function.