The values of x for which the domain of definition of the function, , where [.] denotes the greatest integer part, is not defined are
The function is defined for all real values of x except those
which satisfy the equation
[|x – 1|] + [|7 – x|] – 6 = 0 …………(1)
Case I: (1 < x < 7)
Equation (1) reduces to
[x – 1] + [7 – x] – 6 = 0
i.e., [x] – 1 + [– x] + 7 – 6 = 0 or [x] + [– x] = 0
which is true ∀ x ∈ I
Thus, every integer in (1, 7) satisfies equation (1).
Case II: (x ≤ 1)
Equation (1) reduces to
[1 – x] + [7 – x] – 6 = 0
i.e., 1 + [– x] + 7 + [– x] – 6 = 0
i.e., [– x] = – 1 i.e., – 1 ≤ – x < 0 or 0 < x ≤ 1
Thus, equation (1) is satisfied ∀ 0 < x ≤ 1
Case III: (x ≥ 7)
Equation (1) reduces to
[x – 1] + [x – 7] – 6 = 0
i.e., [x] – 1 + [x] – 7 – 6 = 0 or [x] = 7
7 ≤ x < 8
Thus, equation (1) is satisfied ∀ 7 ≤ x < 8. The union of the
intervals obtained in the above three cases gives the domain
of definition as
R – (0, 1] – [7, 8) – {2, 3, 4, 5, 6}
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