Reduce two variables to a ratio

Question

I was playing around with these two functions : $$ f_1(x)=\frac{a}{ax+b} \\f_2(x)=\frac{\sqrt{a}\sqrt{a+b}}{ax+b} $$ and I realized that I can rewrite $f_1(x)$ as $$ f_1(x)=\frac{c}{cx+1} \quad \text{where }c=\frac{a}{b} $$ This left me wondering if I could do the same with $f_2(x)$ but I don't know how to approach this kind of problem.

Answer 1

Here's a hint

$$ \frac{\sqrt{a}\sqrt{a+b}}{ax + b} = \frac{\sqrt{a^2+ab}}{ax + b} $$

as

$$ \sqrt{a}\cdot\sqrt{b} = \sqrt{a \cdot b}$$

now what would happen if you divide both the numerator and denominator by b?

$$ \frac{\sqrt{a^2+ab}}{ax + b} = \frac{\frac{\sqrt{a^2+ab}}{\sqrt{b^2}}}{\frac{ax + b}{b}} = \frac{\sqrt{\frac{a^2 + ab}{b^2}}}{\frac{a}{b}x + \frac{b}{b}} = \frac{\sqrt{\frac{a^2}{b^2}+\frac{a}{b}}}{\frac{a}{b}x + 1} = \frac{\sqrt{c^2 + c}}{cx + 1} $$