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!