Domain of cos ^{-1}left[log _5left(x^2+7 x+15 | vedclass

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(D) The function $f(x) = \cos ^{-1}\left[\log _5\left(x^2+7 x+15\right)\right]$ is defined when the argument of $\cos ^{-1}$ lies in the interval $[-1

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$[-5,-2]$

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(D) The function $f(x) = \cos ^{-1}\left[\log _5\left(x^2+7 x+15\right)\right]$ is defined when the argument of $\cos ^{-1}$ lies in the interval $[-1, 1]$.So, $-1 \leq \log _5\left(x^2+7 x+15\right) \leq 1$.Applying the base $5$ exponential function, we get $5^{-1} \leq x^2+7 x+15 \leq 5^1$, which is $\frac{1}{5} \leq x^2+7 x+15 \leq 5$.First, consider $x^2+7 x+15 \leq 5$, which implies $x^2+7 x+10 \leq 0$.Factoring the quadratic, we get $(x+5)(x+2) \leq 0$, which gives $x \in [-5, -2]$.Next, consider $x^2+7 x+15 \geq \frac{1}{5}$, which implies $x^2+7 x + 14.8 \geq 0$.The discriminant of $x^2+7 x + 14.8$ is $D = 7^2 - 4(1)(14.8) = 49 - 59.2 = -10.2 Since the leading coefficient is positive and $D Thus, the domain is the intersection of $x \in [-5, -2]$ and $x \in \mathbb{R}$, which is $x \in [-5, -2]$.

Domain of $\cos ^{-1}\left[\log _5\left(x^2+7 x+15\right)\right]$ is

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