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Question
Determine the values of p and q such that the following function is continuous on the entire real number line.
f(x) `{:(= x + 1",", "for" 1 < x < 3),(= x^2 + "p"x + "q"",", "for" |x - 2| ≥ 1):}`
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Solution
|x – 2| = ± (x – 2) and |x – 2| ≥ 1
∴ x – 2 ≥ 1 or 2 – x ≥ 1
∴ x ≥ 3 or 1 ≥ x i.e., x ≤ 1
∴ the given function is
f(x) `{:(= x^2 + "p"x + "q","," "for" x ≤ 1),(= x + 1, "," "for" 1 < x < 3),(= x^2 + "p"x + "q", "," "for" x ≥ 3):}`
If f is continuous on the entire number line, then it is continuous at x = 1 and x = 3
Continuity at x = 1
Since f is continuous at x = 1,
`lim_(x -> 1^+) "f"(x) = "f"(1)`
∴ `lim_(x -> 1) (x + 1) = (x^2 + "p"x + "q")`
∴ 1 + 1 = 1 + p + q
∴ p + q = 1 ...(1)
Continuity at x = 3
Since f is continuous at x = 3,
`lim_(x -> 3^-) "f"(x)` = f(3)
∴ `lim_(x -> 3) (x + 1) = (x^2 + "p"x + "q")`
∴ 3 + 1 = 9 + 3p + q
∴ 3p + q = – 5
∴ 3p + (1 – p) = – 5 ...[By (1)]
∴ 3p - p = - 5 - 1
∴ 2p = – 6
∴ p = – 3
Substituting p = – 3 in (1), we get,
∴ – 3 + q = 1
∴ q = 1 + 3
∴ q = 4
Hence, p = – 3, q = 4.
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