For the following two (02) items: Consider the function f(x) =...

Given $f(x) = \dfrac{x}{1 - x}$

Substitute $\sqrt{x}$ in f(x):

$f(\sqrt{x}) = \dfrac{\sqrt{x}}{1 - \sqrt{x}}$

Substitute $\sqrt{x} + 1$ in f(x):

$f(\sqrt{x} + 1) = \dfrac{\sqrt{x} + 1}{1 - (\sqrt{x} + 1)} = \dfrac{\sqrt{x} + 1}{-\sqrt{x}}$

Now, evaluate the given expression:

$(1 - x)f(\sqrt{x}) + xf(\sqrt{x} + 1) = (1 - x) \left(\dfrac{\sqrt{x}}{1 - \sqrt{x}}\right) +$ x\left(\dfrac{\sqrt{x} + 1}{-\sqrt{x}}\right)

Using 1 - x = (1 - \sqrt{x})(1 + \sqrt{x}):

$= (1 -$\sqrt{x})(1 + \sqrt{x})\left(\dfrac{\sqrt{x}}{1 - \sqrt{x}}\right) - \sqrt{x}(\sqrt{x} +1)$<br><br>$= \sqrt{x}(1 + \sqrt{x}) - \sqrt{x}(\sqrt{x} +1)$<br><br>$= 0$<br><br>Answer:$0