\section*{Exercise 2 - 041232}

Let

\[
f(x)=\frac{a x+b}{c x+d}
\]

with real numbers \(a, b, c, d\) as coefficients \((c \neq 0)\).

For which real numbers \(\mathrm{x}\) is the function defined by the mapping \(x \rightarrow y=f(x)\)?

Without using differential calculus, specify the conditions that the coefficients \(a, b, c, d\) must satisfy for this function to be strictly monotonically decreasing in each of its domains.
}

The first question is somewhat peculiarly formulated. It is presumably asking for the maximal domain within the real numbers.

The denominator must not be 0, so \(c x \neq -d\) or \(x \neq -\frac{d}{c}\) must hold \((c \neq 0\) !).

For all other real \(x\), the numerator is defined and the denominator is not zero, so the quotient \(\frac{a x + b}{c x + d}\) is defined.

The maximal domain is thus \(D = \mathbb{R} \backslash \left\{-\frac{d}{c}\right\}\).

It holds that

\[
a x + b = \frac{a}{c} \cdot (c x + d) + \left(b - \frac{a d}{c}\right)
\]

Therefore, \(f(x)\) can be rewritten as

\[
f(x) = \frac{a x + b}{c x + d} = \frac{\frac{a}{c} \cdot (c x + d)}{c x + d} + \frac{b - \frac{a d}{c}}{c x + d} = \frac{a}{c} + \frac{b - \frac{a d}{c}}{c x + d}
\]

Let \(x_{1}\) and \(x_{2}\) be arbitrary real numbers with \(x_{1} < x_{2}\). Then

\[
f(x_{1}) > f(x_{2}) \Leftrightarrow \frac{a}{c} + \frac{b - \frac{a d}{c}}{c x_{1} + d} > \frac{a}{c} + \frac{b - \frac{a d}{c}}{c x_{2} + d} \Leftrightarrow \frac{b - \frac{a d}{c}}{c x_{1} + d} > \frac{b - \frac{a d}{c}}{c x_{2} + d}
\]

Multiplication by \((c x_{1} + d)(c x_{2} + d)\), which is positive in both cases, yields:

\[
\begin{aligned}
& f(x_{1}) > f(x_{2}) \Leftrightarrow \left(b - \frac{a d}{c}\right) \cdot (c x_{2} + d) > \left(b - \frac{a d}{c}\right) \cdot (c x_{1} + d) \Leftrightarrow \\
& \Leftrightarrow \left(b - \frac{a d}{c}\right) \cdot c \cdot (x_{2} - x_{1}) > 0 \Leftrightarrow (b c - a d) > 0 \quad \left(\text{since } x_{2} - x_{1} > 0\right)
\end{aligned}
\]

The function \(f\) is thus strictly monotonically decreasing in the two intervals \(\left(-\infty, -\frac{d}{c}\right)\) and \(\left(-\frac{d}{c}, \infty\right)\) if and only if \(b c > a d\) holds.