To assist with this problem, let’s analyze the function given in the image:

$f(x) = \frac{3}{4x^2 - 6}$

Step-by-Step Analysis:

1. Determine x-intercepts:

   • x-intercepts occur where $f(x) = 0$.
   • Since $f(x) = \frac{3}{4x^2 - 6}$, the numerator is a constant (3), which means there are no values of $x$ that will make $f(x) = 0$.
   • Therefore, there is no x-intercept.

2. Determine y-intercept:

   • y-intercept occurs where $x = 0$.
   • Substituting $x = 0$ into $f(x)$: $f(0) = \frac{3}{4(0)^2 - 6} = \frac{3}{-6} = -\frac{1}{2}$
   • The y-intercept is at $(0, -\frac{1}{2})$.

3. Vertical Asymptotes:

   • Vertical asymptotes occur where the denominator equals zero (and the numerator is non-zero).
   • Setting $4x^2 - 6 = 0$: $4x^2 = 6 \Rightarrow x^2 = \frac{3}{2} \Rightarrow x = \pm \sqrt{\frac{3}{2}}$
   • This simplifies to $x = \pm \sqrt{\frac{3}{2}} \approx \pm 1.22$.
   • Therefore, there are vertical asymptotes at $x = \pm \sqrt{\frac{3}{2}}$.

4. Horizontal Asymptote:

   • To determine the horizontal asymptote, analyze the degrees of the numerator and denominator.
   • Here, the degree of the numerator is 0 (constant term), and the degree of the denominator is 2.
   • Since the degree of the denominator is higher than the degree of the numerator, the horizontal asymptote is at $y = 0$.

5. Holes:

   • Holes occur where there is a common factor in the numerator and denominator, but since the numerator is a constant and does not share any factors with the denominator, there are no holes in this function.

Summary of Findings:

• x-intercept: DNE (does not exist).
• y-intercept: $(0, -\frac{1}{2})$.
• Vertical asymptotes: $x = \pm \sqrt{\frac{3}{2}}$.
• Horizontal asymptote: $y = 0$.
• Holes: DNE (no holes).

Would you like further details on any of these steps?

Here are some related questions for additional exploration:

1. How would the graph change if the numerator were a polynomial rather than a constant?
2. What is the behavior of the function as $x$ approaches each vertical asymptote?
3. How would you find oblique asymptotes if the degrees of the numerator and denominator were different?
4. What are the implications of having no x-intercepts for a rational function?
5. How would this analysis change if there were additional terms in the denominator?

Tip: Always check the degrees of the numerator and denominator first when analyzing asymptotes for rational functions!