Question

1. write a function that has a zero at x = 2 and a hole at x = - 2.

Explanation:

Step1: Recall zero - factor relationship

A zero at \(x = 2\) means \((x - 2)\) is a factor of the numerator.

Step2: Recall hole - factor relationship

A hole at \(x=- 2\) means \((x + 2)\) is a common factor of the numerator and the denominator.

Step3: Construct the function

We can write the rational function \(f(x)=\frac{(x - 2)(x + 2)}{(x + 2)}\). Simplifying for \(x
eq - 2\), it has the desired properties.

Answer:

\(f(x)=\frac{(x - 2)(x + 2)}{(x + 2)}\) (or any non - zero multiple of this form)