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We have,
Clearly, f(x) assumes real values for all real values for all x except for the values of x = 0
Hence, Domain (f) = R - {0}
Clearly, f(x) assumes real values for all real values for all x except for the values of x satisfying x - 7 = 0 i.e., x = 7
Hence, Domain (f)= R - {7}
We observe that f(x) is a rational function of x asis a rational expression. Clearly, f(x) assumes real values for all x except for the values of x for which x + 1 = 0 i.e., x = -1
Hence, Domain = R - {-1}
We observe that f(x) is a rational function of x asis a rational expression. Clearly, f(x) assumes real values for all x except for all those values of x for which x2 - 9 = 0 i.e., x = -3, 3
Hence, Domain (f) = R - {-3, 3}.
Clearly, f(x) is a rational function of x asis a rational expression in x. We observe that f(x) assumes real values for all x except for all those values of x for which x2 - 8x + 12 = 0 i.e., x = 2, 6
∴ Domain (f) = R - {2, 6}
(i) We have,
Clearly, f(x) assumes real values, if
x - 2 ≥ 0
⟹ x ≥ 2
⟹ x ϵ [2, ∞)
Hence, Domain (f) = [2,∞]
(ii) We have,
x2 - 1 > 0
⟹ (x - 1)(x + 1) > 0 [∵ a2 - b2 = (a - b)(a + b)]
⟹ x < -1 or x > 1
⟹ x ϵ (-∞,-1) ∪ (1,∞)
Hence, domain (f) = (-∞, -1) ∪ (1,∞)
(iii) We have,
9 - x2 ≥ 0
⟹ 9 ≥ x2
⟹ x2 ≤ 9
⟹ -3 ≤ x ≤ 3
⟹ x ϵ [-3, 3]
Hence, domain (f) = [-3, 3]
(iv) We have,
x - 2 ≥ 0 and 3 - x > 0
⟹ x ≥ 2 and 3 > x
⟹ x ϵ [2, 3]
Hence, domain (f) = [2, 3).
We observe that f(x) is a rational function of x asis a rational expression. Clearly, f(x) assumes real values for all x except for the values of x for which
bx - a = 0 i a., bx = a
Clearly, x will take real value for all x ϵ R except for
by - a = 0
⟹ by = a
We observe that f(x) is a rational function of x asis a rational expression. Clearly, f(x) assumes real values for all x except for all those values of x for which cx - d = 0 i.e., cx = d
Clearly, x assumes real values for all y except
x - 1 ≥ 0
⟹ x ≥ 1
⟹ x ϵ [1, ∞)
Hence, domain(f) = [1,∞)
Range: For x ≥ 1, we have,
⟹ f(x) ≥ 0
Thus, f(x) takes all real values greater than zero.
Hence, range(f) = [0, ∞)
Clearly, f(x) assumes values, if
x - 3 ≥ 0
⟹ x ≥ 3
⟹ x ϵ [3,∞)
Hence, domain(f) = [3,∞)
Range: For x ≥ 3,we have,
⟹ x - 3 ≥ 0
⟹ f(x) ≥ 3
Hence, range(f) = [0,∞)
Domain of f : Clearly, f(x) is defined for all x ϵ R except for which
2 - x ≠ 0 i.e., x ≠ 2
Hence, domain (f) = R - (2)
Range of f : Let f(x) = y
f(x)= |x - 1|
Clearly, f(x) is defined for all x ϵ R
⟹ Domain (f) = R
Range: Let f(x) = y
⟹ |x - 1|= y
⟹ f(x) ≥ 0 ∀ × ϵ R
It follows from the above relation that y takes all real values greater or equal to zero.
∴ Range (f) = [0, ∞)
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Chapter 3: Functions – Exercise 3.2...
Chapter 3: Functions – Exercise 3.1...
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