$a,b,c,d:=$non zero real number constants.
$t\geq0$
$f(t):=\frac{a+bt}{c+dt}$
The thing which I want to do is determine the values of the above $4$ constants to make $f(t)$ as a constant function.
The textbook gave the condition for it as below.
$\frac{a}{c}=\frac{b}{d}$
Why this equation is able to be said as the condition for composing a constant function?
On
We want $ f(t) = \frac{a + bt}{c + dt} = e$, where $e$ is some constant
This should hold for all values of t. So taking $t = 0$
$ \frac{a}{c} = e \Rightarrow a = ec$
If $t = 1$:
$\frac{a+b}{c+d} = e \Rightarrow a + b = (c + d)e = ec + ed = a + ed$
$\Rightarrow b = ed \Rightarrow e = \frac{b}{d}$
$\Rightarrow \frac{a}{c} = \frac{b}{d}$
Taking $t=0$ and $t=1$, we obtain that $\frac{a}{c} = \frac{b}{d}$ is a necessary condition. Conversely, if $a = \frac{bc}{d}$ then $\frac{bc/d + bt}{c+dt} = \frac{b}{d}$, a constant.