To find the points of intersection with the x-axis for a function, we need to determine where the function $F(x)$ equals zero. This occurs when the numerator of the function is zero (since a fraction is zero when its numerator is zero, provided the denominator is not zero).

Given the function:

$F(x) = \frac{4x^2 - 3}{2x^2 + 1}$

Step-by-step Solution:

1. Set the numerator equal to zero:
   $4x^2 - 3 = 0$

2. Solve for $x$:
   $4x^2 = 3$
   $x^2 = \frac{3}{4}$
   $x = \pm \sqrt{\frac{3}{4}}$
   $x = \pm \frac{\sqrt{3}}{2}$

3. Verify that the denominator is not zero:
   The denominator is: $2x^2 + 1$
   Plugging in $x = \pm \frac{\sqrt{3}}{2}$:

   $2\left(\frac{\sqrt{3}}{2}\right)^2 + 1 = 2 \times \frac{3}{4} + 1 = \frac{3}{2} + 1 = \frac{5}{2} \neq 0$

   Thus, the denominator is not zero for these values of $x$.

Conclusion:

The points of intersection with the x-axis are:

$\left( \frac{\sqrt{3}}{2}, 0 \right) \quad \text{and} \quad \left( -\frac{\sqrt{3}}{2}, 0 \right)$

Would you like more details or have any questions?

Related Questions:

1. How do you find the intersection points with the y-axis for a rational function?
2. What happens if both the numerator and denominator are zero at a point?
3. How do you find vertical asymptotes for a rational function?
4. What is the significance of holes in the graph of a rational function?
5. How can you determine if a function is continuous at a given point?

Tip:

For rational functions, always check both the numerator and the denominator when looking for zeros or intersections to avoid division by zero issues.