Construction of the function given its properties

Problem

For each of the following sets of requirements name a function satisfying them.

1. The function \(g\) has a maximum domain given by \(]-\infty; 5]\).

2. The function \(k\) has a zero at \(x=2\) as well as pole at \(x=-3\) without changing its sign. The graph of \(k\) has an asymptote the straight line given by \(y=1\) als Asymptote.

Solution of part a

A function with domain \([0; \infty[\) is given by \(x\mapsto\sqrt{x}\). Therefore, \(g(x)=\sqrt{5-x}\) is one of the functions with the given domain.

Solution of part b

The function \(k(x)\) can be chosen as rational function. Beacuase of the zero at \(x=2\) the numerator must contain at least a factor \(x-2\). The pole at \(x=-3\) without change of sign is obtained by means of a factor \((x+3)^2\) in the denominator. In order to obtain the desired asymptotic behavior for \(\vert x\vert\to\infty\), the factor in the numerator must be squares. We thus arrive at

\[k(x)=\frac{(x-2)^2}{(x+3)^2}.\]

We demonstrate with the help of Sage that this function indeed has the required properties.

Zero at \(x=2\):

Pole at \(x=-3\) without change of sign:

Asymptotic approach to the straight line \(y=1\) for \(\vert x\vert\to 1\):