Page 13 - mathsvol1ch1to3ans

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a) f −1 (x) = b) f −1 (x) =
 
x x < 1 −x x < 1
√ √


x 1 ≤ x ≤ 16 x 1 ≤ x ≤ 16
x 2 x 2


x > 16 x > 16
 
64 64
c) f −1 (x) = d) f −1 (x) =
 
x 2 x < 1 2x x < 1
√ √


x 1 ≤ x ≤ 16 x 1 ≤ x ≤ 16
x 2 x 2


x > 16 x > 16
 
16 8
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23. Let f : R → R be defined by f(x) = 1 − |x|. Then the range of f is
(a) R (b) (1, ∞) (c) (−1, ∞) (d) (−∞, 1]
Answer: (d) (−∞, 1]
24. The function f : R → R is defined by f(x) = sinx + cosx is
(a) an odd function (b) neither an odd function nor an even func-
tion
(c) an even function (d) both odd function and even function.
Answer:(b) neither an odd function nor an even function
Explanation: f(−x) = − sin x + cos x
Neither f(−x) = f(x) nor f(−x) = −f(x)
4
2
(x + cosx)(1 + x )
25. The function f : R → R is defined by f(x) = + e −|x| is
3
(x − sinx)(2x − x )
(a)an odd function (b)neither an odd function nor an even function
(c)an even function (d)both odd function and even function.
Answer:(c) an even function
Explanation
4
2
(−x) + cos − x)(1 + (−x) )
f(−x) = + e −|−x|
((−x) − sin(−x))(2(−x) − (−x) )
3
4
2
(x + cosx)(1 + x )
f(−x) = + e −|x|
3
−(x − sinx)(−1)(2x − x )
Hence f(−x) = f(x).
Exercise - 2.6
√
1. Classify each element of { 7, −1 , 0, 3.14, 4, 22 } as a member of
4 7
(i) N (ii) Q (iii) R − Q (iv) Z.
N Q R − Q Z
Solution: √
,
4 0, −1 22 7 0, 3.14
4
7
√
2. Prove that 3 is an irrational number.
√ √ m
Suppose that 3 is a rational number. Let 3 = , where m and n are positive integers with no
n
2
2
2
common factors greater than 1. Then, we have m = 3n , which implies that m is a multiple of
2
2
2
3 and hence m is a multiple of 3. Let m = 3k. Then, we have 3n = 9k which gives n = 3k 2
Thus, n is also multiple of 3. It follows, that m and n have a common factor 3. Thus, we arrived
√
at a contradiction. Hence, 3 is an irrational number.
3. Are there two distinct irrational numbers such that their difference is a rational number? Justify.
Solution: We can find suitable irrational numbers such that their difference is a rational number.
√
√
For example 2 + 1 is an irrational number, then 2 − 1 is also an irrational number. But the
difference is 2 which is a rational number.