Final answer:
The question asks for the creation of a rational function with given features including vertical asymptote, zeroes, and a hole. One possible function that satisfies this is f(x) = x(x-3)(x-5)/(x-4)(x-5), with the appropriate characteristics.
Step-by-step explanation:
The student is asking about creating a rational function based on given characteristics: a vertical asymptote at x=4, zeroes at x=0 and x=3, a hole when x=5, and no horizontal asymptotes. To construct such a function, we can utilize these attributes as follows:
- A vertical asymptote at x=4 suggests that there is a factor in the denominator that becomes zero at x=4. Therefore, the denominator could be (x-4).
- Zeroes at x=0 and x=3 indicate that x and (x-3) are factors in the numerator, since setting the numerator equal to zero should give us these x-values.
- A hole at x=5 is created when there is a factor in both the numerator and the denominator that cancel out. Hence, (x-5) should be present in both the numerator and the denominator.
Combining this information, one possible function that satisfies all these conditions is:
f(x) = \frac{x(x-3)(x-5)}{(x-4)(x-5)}
This function has the required zeroes and vertical asymptote, a removable discontinuity (hole) at x=5, and since the degrees of the numerator and denominator are equal, there is no horizontal asymptote.