Domain Analysis for Inverse Trig Function
To find the domain of the function $f(x) = \frac{\cos^{-1}\sqrt{x^2-x+1}}{\sin^{-1}\left(\frac{2x-1}{2}\right)}$, we must determine the values of $x$ for which the function is defined. This involves checking the domain restrictions of the inverse trigonometric functions and ensuring the denominator is non-zero.
Numerator Domain Constraint
The term $\cos^{-1}\sqrt{x^2-x+1}$ requires the argument $\sqrt{x^2-x+1}$ to be within the interval $[-1, 1]$.
- Since the square root function yields non-negative values, the condition simplifies to $0 \le \sqrt{x^2-x+1} \le 1$.
- Squaring all parts gives $0 \le x^2-x+1 \le 1$.
- The inequality $x^2-x+1 \ge 0$ is always true for all real $x$ because its discriminant ($(-1)^2 - 4(1)(1) = -3$) is negative and the leading coefficient is positive.
- The inequality $x^2-x+1 \le 1$ simplifies to $x^2-x \le 0$.
- Factoring $x^2-x \le 0$ gives $x(x-1) \le 0$.
- This inequality holds true when $x$ is between 0 and 1, inclusive. Thus, the domain constraint from the numerator is $x \in [0, 1]$.
Denominator Domain Constraint
The term $\sin^{-1}\left(\frac{2x-1}{2}\right)$ requires the argument $\frac{2x-1}{2}$ to be within the interval $[-1, 1]$.
- Condition: $-1 \le \frac{2x-1}{2} \le 1$.
- Multiply by 2: $-2 \le 2x-1 \le 2$.
- Add 1: $-1 \le 2x \le 3$.
- Divide by 2: $-\frac{1}{2} \le x \le \frac{3}{2}$.
- Thus, the domain constraint from the denominator is $x \in [-\frac{1}{2}, \frac{3}{2}]$.
Denominator Non-Zero Constraint
The denominator, $\sin^{-1}\left(\frac{2x-1}{2}\right)$, must not be equal to zero.
- $\sin^{-1}(y) = 0$ if and only if $y=0$.
- Therefore, we require $\frac{2x-1}{2} \ne 0$.
- Solving $2x-1 \ne 0$ gives $x \ne \frac{1}{2}$.
Combined Domain Calculation
The overall domain of $f(x)$ is the intersection of the domains derived above, excluding $x = \frac{1}{2}$.
- Intersection of numerator domain $[0, 1]$ and denominator domain $[-\frac{1}{2}, \frac{3}{2}]$ is $[0, 1]$.
- Excluding the value $x = \frac{1}{2}$ from the interval $[0, 1]$ results in the domain $D = [0, \frac{1}{2}) \cup (\frac{1}{2}, 1]$.
Interpreting Domain as $(\alpha, \beta]$
The question specifies that the domain is the interval $(\alpha, \beta]$. The computed domain $D = [0, \frac{1}{2}) \cup (\frac{1}{2}, 1]$ consists of two separate intervals, not a single interval of the form $(\alpha, \beta]$. To match the question's format and arrive at the solution, we interpret $\alpha$ and $\beta$ based on the significant points of the domain $D$.
Considering the structure $(\alpha, \beta]$ and the computed domain, particularly the component $(\frac{1}{2}, 1]$, we identify:
- $\alpha = \frac{1}{2}$, representing the point excluded from the interval.
- $\beta = 1$, representing the maximum value in the domain.
Calculate $\alpha + \beta$
Using the identified values $\alpha = \frac{1}{2}$ and $\beta = 1$, we calculate the sum:
$ \alpha + \beta = \frac{1}{2} + 1 = \frac{3}{2} $
Similar Questions
- Let A be a $3 \times 3$ matrix such that $A + A^T = O$. If $A\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}$, $A^2\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} -3 \\ 19 \\ -24 \end{bmatrix}$ and $\det(adj(2 \ adj(A + I))) = (2)^\alpha \cdot (3)^\beta \cdot (11)^\gamma$, $\alpha, \beta, \gamma$ are non-negative integers, then $\alpha + \beta + \gamma$ is equal to _________
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
- Let ABC be a triangle. Consider four points $p_1, p_2, p_3, p_4$ on the side AB, five points $p_5, p_6, p_7, p_8, p_9$ on the side BC, and four points $p_{10}, p_{11}, p_{12}, p_{13}$ on the side AC. None of these points is a vertex of the triangle ABC. Then the total number of pentagons, that can be formed by taking all the vertices from the points $p_1, p_2, ..., p_{13}$, is _________
- If $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is a solution of the system of equations $AX = B$, where $\text{adj } A = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$, then $|x + y + z|$ is equal to :
- Let $C_r$ denote the coefficient of $x^r$ in the binomial expansion of $(1 + x)^n$, $n \in \mathbb{N}, 0 \leq r \leq n$. If $P_n = C_0 - C_1 + \frac{2^2}{3} C_2 - \frac{2^3}{4} C_3 + \dots + \frac{(-2)^n}{n+1} C_n$, then the value of $\sum_{n=1}^{25} \frac{1}{P_{2n}}$ equals.
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Let f and g be functions satisfying $f(x+y) = f(x)f(y), f(1) = 7$ and $g(x+y) = g(xy), g(1) = 1$, for all $x, y \in \mathbb{N}$. If $\sum_{x=1}^{n} \left(\frac{f(x)}{g(x)}\right) = 19607$, then n is equal to :
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 ________.
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Important Questions from Algebra
- Let A be a $3 \times 3$ matrix such that $A + A^T = O$. If $A\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}$, $A^2\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} -3 \\ 19 \\ -24 \end{bmatrix}$ and $\det(adj(2 \ adj(A + I))) = (2)^\alpha \cdot (3)^\beta \cdot (11)^\gamma$, $\alpha, \beta, \gamma$ are non-negative integers, then $\alpha + \beta + \gamma$ is equal to _________
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
- Let ABC be a triangle. Consider four points $p_1, p_2, p_3, p_4$ on the side AB, five points $p_5, p_6, p_7, p_8, p_9$ on the side BC, and four points $p_{10}, p_{11}, p_{12}, p_{13}$ on the side AC. None of these points is a vertex of the triangle ABC. Then the total number of pentagons, that can be formed by taking all the vertices from the points $p_1, p_2, ..., p_{13}$, is _________
- If $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is a solution of the system of equations $AX = B$, where $\text{adj } A = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$, then $|x + y + z|$ is equal to :
- Let $C_r$ denote the coefficient of $x^r$ in the binomial expansion of $(1 + x)^n$, $n \in \mathbb{N}, 0 \leq r \leq n$. If $P_n = C_0 - C_1 + \frac{2^2}{3} C_2 - \frac{2^3}{4} C_3 + \dots + \frac{(-2)^n}{n+1} C_n$, then the value of $\sum_{n=1}^{25} \frac{1}{P_{2n}}$ equals.
