91Ó°ÊÓ

The discriminant is a useful tool in algebra that helps us determine the nature of the roots of a quadratic equation. For any quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \( D \) is calculated using the formula \(D = b^2 - 4ac\). This value tells us:

• If \, \(D > 0\): The quadratic equation has two distinct real roots.
• If \, \(D = 0\): There is exactly one real root, known as a double or repeated root.
• If \, \(D < 0\): The equation has no real roots, implying the roots are complex or imaginary numbers.

In the exercise, the discriminants of both the numerator \(2x^2 + 3x + 4\) and the denominator \(x^2 + 4x + 7\) were calculated to be less than zero. This means neither the numerator nor the denominator have real roots, pointing out that the rational function does not become zero or undefined at any real point.

A function is considered continuous if it has no breaks, jumps, or holes over its domain. When dealing with rational functions, continuity is specifically crucial at points where the function is defined. For a function \(F(x) = \frac{p(x)}{q(x)}\) to be continuous at a point, the denominator \(q(x)\) should not be zero, as this would cause a division by zero, creating a point of discontinuity.

In this particular exercise, the rational function \(F(x) = \frac{2x^2 + 3x + 4}{x^2 + 4x + 7}\) is examined. Since the quadratic equation for the denominator has no real roots, it does not equal zero for any real number \(x\). Therefore, the function remains continuous over the entire set of real numbers. This continuity is significant because it implies the function maintains a smooth and unbroken graph, which can be analyzed without concern for defining boundaries or encountering undefined points.

Analyzing the numerator and denominator of a rational function helps us understand its behavior and critical points. Here the function is given by \(F(x) = \frac{2x^2 + 3x + 4}{x^2 + 4x + 7}\). The first important aspect is:

• Numerator Analysis: To find where the whole function equals zero, solve \(p(x) = 0\), here \(2x^2 + 3x + 4 = 0\). However, as found, this equation has a negative discriminant, suggesting it has no real solutions, meaning the rational function does not reach zero.
• Denominator Analysis: Identify points where the function might be undefined by solving \(q(x) = 0\), for this case \(x^2 + 4x + 7 = 0\). Similarly, this also holds no real solution due to a negative discriminant, indicating the function doesn’t get undefined at any real point.

From this analysis, we can infer that the function is valid for all real numbers \(x\), as neither the numerator nor the denominator result in problematic roots that would disrupt the function's continuity or existence. This type of detailed numerator and denominator analysis assists in understanding how to work with and interpret rational functions effectively.