To find the domains of the functions \(f(x) = \log_4\log_3\log_7 (8-\log_2(x^2+4x+5))\) and \(g(x) = \sin^{-1} \left(\frac{7x + 10}{x-2}\right)\), we need to ensure the arguments of all logarithmic and inverse trigonometric functions are defined.
Step 1: Domain of \(f(x)\)
1. For the innermost function, \(\log_2(x^2+4x+5)\):
- We need \(x^2 + 4x + 5 > 0\). The discriminant \(b^2 - 4ac = 16 - 20 = -4\), which is negative, indicating the quadratic is always positive.
2. The outer function \(8-\log_2(x^2+4x+5)\):
- Requires \(8-\log_2(x^2+4x+5) > 1\), hence \(\log_2(x^2+4x+5) < 8\). Converting to exponential form, \(x^2 + 4x + 5 < 2^8\).
- Simplifying gives \(x^2 + 4x - 251 < 0\), factoring to \((x - 13)(x + 19) < 0\). Using a number line, this holds for \(x \in (-19, 13)\).
3. Finally, for \(\log_4\log_3(y)\) to be defined:
- The argument must be positive: \(\log_3\log_7(8 - \log_2(x^2+4x+5)) > 0\), implying \(8 - \log_2(x^2+4x+5) > 7\), thus \(\log_2(x^2+4x+5) < 1\). Hence, \(x^2 + 4x + 5 < 2\), or (x + 1)(x + 3) < 0.
- The solutions are \(x \in (-3, -1)\).
The domain of \(f(x)\) is \((\alpha, \beta) = (-3, -1)\).
Step 2: Domain of \(g(x)\)
1. For \(\sin^{-1}\left(\frac{7x+10}{x-2}\right)\) to be defined, the argument must be in \([-1, 1]\):
- Hence, \(-1 \leq \frac{7x+10}{x-2} \leq 1\).
- Consider \(\frac{7x+10}{x-2} \geq -1\): This simplifies to \(7x + 10 \geq -x + 2\) or \(8x \geq -8\). So, \(x \geq -1\).
- Consider \(\frac{7x+10}{x-2} \leq 1\): This simplifies to \(7x + 10 \leq x - 2\) or \(6x \leq -12\), giving \(x \leq -2\).
Hence, intersection of both gives \(x \in [-2, -1]\).
The domain of \(g(x)\) is \([\gamma, \delta] = [-2, -1]\).
Final Calculation
Calculate \(\alpha^2 + \beta^2 + \gamma^2 + \delta^2\):
- \(\alpha = -3, \beta = -1, \gamma = -2, \delta = -1\)
- \(\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = (-3)^2 + (-1)^2 + (-2)^2 + (-1)^2 = 9 + 1 + 4 + 1 = 15\)
Thus, the final answer is 15.
Similar Questions
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
- Let $a_1 = 1$ and for $n \ge 1, a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2(n+1)^2}$. Then $\left| \sum_{n=1}^\infty \left( a_n - \frac{2}{n^2} \right) \right|$ is equal to ____________.
If $ \left(\frac{1}{^{15}C_0} + \frac{1}{^{15}C_1}\right)$ $\left(\frac{1}{^{15}C_1} + \frac{1}{^{15}C_2}\right)$ $\dots$ $ \left(\frac{1}{^{15}C_{12}} + \frac{1}{^{15}C_{13}}\right)$ = $\frac{\alpha^{13}}{^{14}C_0 ^{14}C_1 \dots ^{14}C_{12}} $, then $30\alpha$ is equal to __________.
- Three persons enter in a lift at the ground floor. The lift will go upto 10$^{\text{th}}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to _________.
- The number of 4-letter words, with or without meaning, which can be formed using the letters PQR PQRSTUVP, is ___________.
- Let $|A| = 6$, where A is a $3 \times 3$ matrix. If $|\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))| = 2^m \cdot 3^n$, $m, n \in \mathbf{N}$, then $m + n$ is equal to ___________.
Important Questions from Algebra
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
