X = a/ (1+a) . Would the value of x increase or decrease if “a” increases?

Question

It’s pretty easy to find the answer after putting in random values but I can’t wrap my head around with my below explanation which is in direct contradiction to the result.

If the numerator increases the x value goes up and the value of x goes down if the denominator increases. Now if “a” increases the net increase of the denominator would be more than the net increase of numerator as the denominator has “+1” in it. Therefore, since the denominator is more than that of numerator shouldn’t the value of x decrease with increase of “a”?

Answer 1

Note that $X = \frac{a}{1 + a} = 1 - \frac{1}{1 + a}$. Then, since $\frac{1}{1 + a}$ is a decreasing function, $X$ increases as $a$ increases. $\blacksquare$

Answer 2

Alternative approach:

$x = \frac{a}{1 + a} = \frac{1}{\frac{1}{a} + 1}.$

As $a$ increases, the first term in the denominator decreases.

Therefore, as $a$ increases, the denominator decreases.

Therefore, as $a$ increases, $x$ increases.