If the domain of the function is , then is equal to [2025]

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Solution

Q.
If the domain of the function $f (x) = \log_{7} (1 - \log_{4} (x^{2} - 9 x + 18))$ is $(α, β) \cup (γ, δ)$ , then $α + β + γ + δ$ is equal to [2025]

1 18

2 15

3 16

4 17

Ans.
(1)

For f(x) to be defined we have,

$1 - \log_{4} (x^{2} - 9 x + 18) > 0 i . e ., x^{2} - 9 x + 18 < 4$

Also, $x^{2} - 9 x + 18 > 0$

$\Rightarrow (x - 3) (x - 6) > 0$

$\Rightarrow x \in (- \infty, 3) \cup (6, \infty)$ ... (i)

Now, $x^{2} - 9 x + 18 < 4$

$\Rightarrow x^{2} - 9 x + 14 < 0$

$\Rightarrow (x - 2) (x - 7) < 0$

$\Rightarrow x \in (2, 7)$ ... (ii)

From equation (i) & (ii), we get

$x \in (2, 3) \cup (6, 7) = (α, β) \cup (γ, δ)$ [Given]

Hence, $α + β + γ + δ = 2 + 3 + 6 + 7 = 18$ .

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