https://file.aiquickdraw.com/mathbot/file/1727349188398c4azfgiw.jpg

The expression in the image is:

\[
y = -\sqrt{\frac{x + 1}{x}}
\]

Let's first calculate the **domain** of this function:

### Step 1: Set conditions for the square root
The expression inside the square root must be non-negative, i.e.:

\[
\frac{x + 1}{x} \geq 0
\]

This inequality can be solved by analyzing the critical points where the numerator and denominator are zero:

- The numerator \(x + 1 = 0\) implies \(x = -1\).
- The denominator \(x = 0\) is undefined, so we must exclude \(x = 0\) from the domain.

Now, analyze the sign of the fraction in the intervals determined by these points:

- For \(x < -1\), both \(x + 1 < 0\) and \(x < 0\), so the fraction is positive.
- For \(-1 < x < 0\), \(x + 1 > 0\) but \(x < 0\), so the fraction is negative.
- For \(x > 0\), both \(x + 1 > 0\) and \(x > 0\), so the fraction is positive.

Thus, the inequality holds for \(x \in (-\infty, -1] \cup (0, \infty)\).

### Step 2: Consider the square root
Since we are dealing with a square root, the expression is defined where \(\frac{x+1}{x} \geq 0\), excluding \(x = 0\), as already determined. Therefore, the **domain** of the function is:

\[
\boxed{(-\infty, -1] \cup (0, \infty)}
\]

### Step 3: Intersections with axes

1. **Intersection with the y-axis:**
   To find the intersection with the y-axis, we evaluate the function at \(x = 0\), but since \(x = 0\) is excluded from the domain, there is no intersection with the y-axis.

2. **Intersection with the x-axis:**
   The function will intersect the x-axis when \(y = 0\). Set the function equal to zero:

   \[
   -\sqrt{\frac{x+1}{x}} = 0
   \]

   This implies that the expression inside the square root must be zero:

   \[
   \frac{x+1}{x} = 0
   \]

   Solving this gives \(x + 1 = 0\), or \(x = -1\). Therefore, the intersection with the x-axis is at the point:

   \[
   (-1, 0)
   \]

### Summary:
- **Domain**: \( (-\infty, -1] \cup (0, \infty) \)
- **Intersection with the x-axis**: \( (-1, 0) \)
- **Intersection with the y-axis**: None (since \(x = 0\) is excluded from the domain)

Would you like further details or have any questions?

Here are some related questions to explore:
1. How do we handle square root functions when determining the domain?
2. Why is \(x = 0\) excluded from the domain of this function?
3. What happens when \(x = -1\) and why is it a critical point?
4. How can the function behavior be analyzed as \(x\) approaches infinity?
5. What kind of transformation does the negative sign outside the square root introduce?

**Tip**: When dealing with square roots in functions, always remember that the radicand must be non-negative to define the function properly!

Explore the domain and intersections of the function y = -√((x + 1)/x). Learn how to determine the valid x values for the square root, analyze intersections with the axes, and apply inequality rules to find the function's behavior. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation