Create a rational function that has a hole at x=5, a vertical asymptote at z=-4, a x-int at x=3 and a horizontally asymptote at y=2

hole at x=5

y = a(x-5)/(x-5)

vertical asymptote at x=-4
y = a(x-5) / (x-5)(x+4)

x-int at x=3
y = a(x-5)(x-3) / (x-5)(x+4)

horizontal asymptote at y=2
y = 2(x-5)(x-3) / (x-5)(x+4)

To create a rational function that meets the given criteria, we need to consider each requirement separately and then combine them into a single function.

1. Hole at x=5: A hole occurs when a factor in the denominator of the rational function cancels out with a factor in the numerator. In this case, we want a hole at x=5. Therefore, we can start with the function (x-5) / (x-5).

2. Vertical asymptote at z=-4: A vertical asymptote occurs when a factor in the denominator approaches zero, but the numerator does not. To create a vertical asymptote at z=-4, we should include a factor of (z+4) in the denominator.

3. X-intercept at x=3: To ensure that the function passes through x=3, we can include a factor of (x-3) in the numerator.

4. Horizontally asymptote at y=2: A horizontal asymptote occurs when the degree of the numerator is equal to or less than the degree of the denominator. To achieve a horizontal asymptote at y=2, we can let the degree of the numerator be equal to the degree of the denominator by multiplying both the numerator and denominator by a constant, let's say "a".

Combining all of the above requirements, we can create the rational function as follows:

f(x) = a(x-5)(x-3) / (x-5)(x+4)

The vertical asymptote is at z = -4, the hole is at x = 5, the x-intercept is at x = 3, and the horizontal asymptote is at y = 2.

Note: The constant "a" can be any non-zero real number, as it only affects the vertical scaling of the graph.