Page 133 - DMTH401_REAL ANALYSIS

Unit 9: Functions

9.7 Review Questions Notes

1
1. Show the graph of f : R  R defined by F(x) = ( ) x
2
3
2. Find the period of the function f where f(x) = |sin x|
3. Test which of the following functions with domain and co-domain as R are bounded and
unbounded:
(i) f(x) = tan x

(ii) f(x) = [x]
(iii) f(x) = e x
(iv) f(x) = log x
4. Suppose t : S  R and g : S  R are any bounded functions on S. Prove that f + g and f. g are
also bounded functions on S.
5. If a, b, c, d are real numbers such that
2
2
2
a + b = I, c + d = 1,
2
then show that ac + bd  1.
6. Prove that |a + b + c| |a|+|b|+|c| for all a, b, c,  R.
7. Show that
|a + a + .... + a |  |a |+|a | + .... +|a | for a , a ,.... a,  R.
1 2 n 1 2 n 1 2
8. Find which of the sets in question 8 are bounded below. Write the infemum if it exists.
9. Which of the sets in question 8 are bounded and unbounded.
10. Justify the following statements:
(i) The identity function is an odd function.
(ii) The absolute value function is an even function.

(iii) The greatest integer function is not onto.
(iv) The tangent function is periodic with period .
(v) The function f(x) = |x| for –2  x  3 is bounded.

x
(vi) The function f(x) = e is not bounded
 
(vii) The function f(x) = sin x, for x [ – , ] is monotonically increasing.
2 2
(viii) The function f(x) = cos x for 0  x  is monotonically decreasing.


(ix) The function f(x) = tan x is strictly increasing for x [0, ].
2

2
2x  3x 2

(x) f(x) = is an algebraic function.
3x 2


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