Saved Bookmarks
| 1. |
The domain of the function \( f(x)=\frac{\cos ^{-1}\left(\frac{x^{2}-5 x+6}{x^{2}-9}\right)}{\log _{e}\left(x^{2}-3 x+2\right)} \) is |
|
Answer» \(x^2 - 3x + 2 > 0\;\text{and}\; x^2 - 3x + 2 \ne 1\) ⇒ \( (x - 1) (x - 2) > 0\;\text{and}\; x^2 - 3x + 1 \ne 0\) ⇒ \(x \in (-\infty , 1) \cup (2, \infty)\;\text{and}\; x \ne \frac{3\pm \sqrt5}2\) Also, \(-1 \le \frac{x^2 - 5x + 6}{x^2 - 9} \le 1 \; \text{and}\; x^2 - 9 \ne 0\) ⇒ \(- x^2 + 9 \le x^2 - 5x + 6 \;\text{and}\;x^2 - 5x + 6 \le x^2 - 9\; \text{and}\; x \ne 3 \) & \(-3\) ⇒ \(2x^2 - 5x - 3 \ge 0 \;\text{and}\; -5x + 6 \le -9\) ⇒ \((x - 3) (2x + 1) \ge 0 \; \text{and}\; 5x \ge 15\) ⇒ \(x\in (-\infty, \frac{-1}2]\cup [3, \infty)\;\text{and}\;x\in(3, \infty).\) ∴ Domain of function is \(x\in (-\infty, \frac{-1}2]\cup [3, \infty) - \{-3\}.\) |
|