Question

pause the video, graph the following rational functions using other graphing technology, then fill in the chart using these graphs plus what you learned earlier in the video. \\( y = \frac{x + 5}{x} \\) \\( y = \frac{x^2 - 9}{x^2 + 4x + 4} \\) \\( y = \frac{2x^2 - 4}{x - 3} \\) | | polynomial functions | rational functions | |---|---|---| | sometimes have roots | | | | always have roots | | | | sometimes have a y-intercept | | | | always have a y-intercept | | | | sometimes have a horizontal asymptote | | | | sometimes have a vertical asymptote | | | | sometimes have holes | | | | have different kinds of end behavior | | | multiple-choice question which functions sometimes have holes? - rational - polynominl

Explanation:

Step1: Analyze polynomial functions

Polynomial functions are of the form \(y = a_nx^n+\cdots+a_1x + a_0\). They are continuous everywhere in their domain (all real numbers). Since there are no common factors in the numerator and denominator (as there is no denominator in the non - rational form of a polynomial, unless we consider the polynomial \(y=\frac{P(x)}{1}\)), polynomial functions do not have holes.

Step2: Analyze rational functions

Rational functions are of the form \(y=\frac{f(x)}{g(x)}\), where \(f(x)\) and \(g(x)\) are polynomials. A hole occurs when there is a common factor in the numerator and the denominator. For example, if \(y = \frac{(x - 1)(x+2)}{(x - 1)(x - 3)}\), when \(x
eq1\), \(y=\frac{x + 2}{x-3}\), and there is a hole at \(x = 1\). But if the rational function is in its simplest form (no common factors), there is no hole. So, rational functions sometimes have holes.

Answer:

rational