Write rational function based on requirements

Question

Write a rational function with the following properties.

1. It has an x-intercept at (-2, 0).

2. It has a hole at (4, f(4)) and a vertical asymptote at x = 1.

3. The end behavior is that the limit is 2 for x reaching infinity or negative infinity.

Please describe step-by-step on how to create the equation.

Answer 1

First, rational functions are CONTINUOUS except at zeroes of the denominator; therefore, a rational function cannot have a jump discontinuity (hole).

Since there is a zero at x=-2, the numerator must include the factor (x+2).

The vertical asymptote at x=1 means the denominator contains the factor (x-1).

If there are no other conditions except the behavior at ±∞, then the numerator must contain a factor of 2.

This information should allow you to construct the function except for the required jump discontinuity.