\( 225 \quad y = \sqrt { \frac { 1 } { x ^ { 2 } - 1 } } \)

To analyze the equation \( y = \sqrt{ \frac{1}{x^2 - 1} } \), we can start by determining the domain of the function. 1. **Identify the domain**: The expression inside the square root, \( \frac{1}{x^2 - 1} \), must be non-negative for \( y \) to be real. This means that \( x^2 - 1 \) must be positive (since the denominator cannot be zero). Therefore, we need: \[ x^2 - 1 > 0 \] This inequality can be factored as: \[ (x - 1)(x + 1) > 0 \] The critical points are \( x = -1 \) and \( x = 1 \). We can test the intervals determined by these points: - For \( x < -1 \): Choose \( x = -2 \): \[ (-2 - 1)(-2 + 1) = (-3)(-1) = 3 > 0 \] - For \( -1 < x < 1 \): Choose \( x = 0 \): \[ (0 - 1)(0 + 1) = (-1)(1) = -1 < 0 \] - For \( x > 1 \): Choose \( x = 2 \): \[ (2 - 1)(2 + 1) = (1)(3) = 3 > 0 \] Thus, the solution to the inequality \( (x - 1)(x + 1) > 0 \) is: \[ x < -1 \quad \text{or} \quad x > 1 \] Therefore, the domain of \( y \) is \( (-\infty, -1) \cup (1, \infty) \). 2. **Behavior of the function**: Next, we can analyze the behavior of the function as \( x \) approaches the boundaries of the domain. - As \( x \to -1^- \) (approaching -1 from the left), \( x^2 - 1 \to 0^+ \), so \( y \to \infty \). - As \( x \to -\infty \), \( x^2 - 1 \to \infty \), so \( y \to 0 \). - As \( x \to 1^+ \) (approaching 1 from the right), \( x^2 - 1 \to 0^+ \), so \( y \to \infty \). - As \( x \to \infty \), \( x^2 - 1 \to \infty \), so \( y \to 0 \). 3. **Summary of the function**: The function \( y = \sqrt{ \frac{1}{x^2 - 1} } \) is defined for \( x < -1 \) and \( x > 1 \). It approaches infinity as \( x \) approaches -1 or 1 from the left or right, respectively, and approaches 0 as \( x \) goes to negative or positive infinity. In conclusion, the function has the following characteristics: - **Domain**: \( (-\infty, -1) \cup (1, \infty) \) - **Range**: \( (0, \infty) \) - **Asymptotic behavior**: \( y \to \infty \) as \( x \to -1^- \) or \( x \to 1^+ \), and \( y \to 0 \) as \( x \to -\infty \) or \( x \to \infty \).