y= sqrt(3-х)(х+1)/x

To analyze the function \( y = \frac{\sqrt{(3 - x)(x + 1)}}{x} \), we can explore its domain, critical points, and behavior.

1. Domain

The expression under the square root, \( (3 - x)(x + 1) \), must be non-negative since you cannot take the square root of a negative number.

Step 1: Find when \( (3 - x)(x + 1) \geq 0 \)

To find the critical points, set \( (3 - x)(x + 1) = 0 \):

• \( 3 - x = 0 \) → \( x = 3 \)
• \( x + 1 = 0 \) → \( x = -1 \)

Step 2: Test intervals Now, test the intervals created by the critical points:

• Interval 1: \( x < -1 \) (choose \( x = -2 \)): \((3 - (-2))( -2 + 1) = (5)(-1) < 0\)

• Interval 2: \(-1 < x < 3\) (choose \( x = 0 \)): \((3 - 0)(0 + 1) = (3)(1) > 0\)

• Interval 3: \( x > 3 \) (choose \( x = 4 \)): \((3 - 4)(4 + 1) = (-1)(5) < 0\)

Conclusion for the domain: The function is non-negative and thus defined for \( -1 \leq x \leq 3 \), excluding \( x = 0 \) since it results in division by zero.

Therefore, the domain is \( [-1, 0) \cup (0, 3] \).

2. Behavior and Asymptotes

• As \( x \) approaches \( 0 \) from the left, \( y \rightarrow -\infty \).
• As \( x \) approaches \( 0 \) from the right, it is undefined.
• At \( x = -1 \): \[ y = \frac{\sqrt{(3 - (-1))(-1 + 1)}}{-1} = \frac{\sqrt{(4)(0)}}{-1} = 0 \]
• At \( x = 3 \): \[ y = \frac{\sqrt{(3 - 3)(3 + 1)}}{3} = 0 \]

3. Summary

The function \( y = \frac{\sqrt{(3 - x)(x + 1)}}{x} \) has the following characteristics:

• Domain: \( [-1, 0) \cup (0, 3] \)
• Critical points: \( y(-1) = 0 \) and \( y(3) = 0 \)
• Asymptotic behavior: Approaches negative infinity as \( x \rightarrow 0^- \)

If you need any specific evaluations, graphs, or derivatives, let me know!