Question

The function f is a rational function graphed in the xy-plane. the plynomial in the numerator of f has exactly one real zero at x=3. The polynomial of the denominator of f has exactly two real zeros at both x=3 and x=6. The multiplicities of the zeros at x=3 in the numerator and in the denominator are equal.

find the domain for the graph of f
Describe any holes and/or vertical asymptotes for the graph of f.
Explain how your answer from part b would change if the multiplicities of the zeros al x = 3 in the numerator and denominator were not equal?

Answer 1

The domain for the graph of f would be all real numbers except for the values that make the denominator equal to zero. So, in this case, the domain would be all real numbers except for x = 3 and x = 6, since those are the zeros of the denominator.

Since the numerator has exactly one real zero at x = 3, there would be a hole at that point on the graph of f. This means that the function is undefined at x = 3, but there is still a point where the graph approaches.

As for vertical asymptotes, there would be one at x = 6, since that is a zero of the denominator but not the numerator. The graph of f would approach infinity or negative infinity as x approaches 6 from either side.

If the multiplicities of the zeros at x = 3 in the numerator and denominator were not equal, it would change the nature of the hole and the vertical asymptote. The hole would still be at x = 3, but the vertical asymptote would be different depending on the multiplicities. If the multiplicity in the numerator is greater, there would be no vertical asymptote. If the multiplicity in the denominator is greater, there would be a vertical asymptote at x = 3.