To find the points of discontinuity for the rational function y = (8x^2 + 5x + 6), we need to analyze the function’s denominator. A rational function can be expressed in the form:
y = f(x) = \frac{P(x)}{Q(x)},
where P(x) is the numerator and Q(x) is the denominator. The points of discontinuity occur when the denominator Q(x) is equal to zero because the function becomes undefined at those x-values.
However, in this case, the expression 8x^2 + 5x + 6 is a quadratic function and does not have a denominator. Therefore, we must check if this quadratic can be zero.
To find the roots of the quadratic, we can apply the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a, where a = 8, b = 5, and c = 6.
Calculating the discriminant:
D = b² – 4ac = 5² – 4 * 8 * 6 = 25 – 192 = -167
Since the discriminant (D) is negative, it indicates that the quadratic has no real roots and, therefore, does not cross the x-axis. This implies that the function y = (8x^2 + 5x + 6) is continuous for all real numbers.
In summary, there are no points of discontinuity for the given rational function.