Answer
$1.\ \ \mathrm{Length\:of\:side\:as\:a\:function\:of\:diagonal:}\ \ a(d)=\frac{d\sqrt{2}}{2}$
$2.\ \ \mathrm{Area\:as\:a\:function\:of\:diagonal:}\ \ A(d)=\frac{d^2}{2}$
Work Step by Step
$1.$ Let's say, $\ a\ $ is the side length of the square $\square ABCD$. If we cut the square into half, we will get two isosceles triangles. Consider the triangle $\triangle\mathrm{BCD}$.
In the considered triangle, let's suppose $\ d\ $ represents the length of diagonal. To represent the length of a side as a function of a diagonal, we need to apply the Pythagorean Theorem as:
$d^2=a^2+a^2$
$\Rightarrow\ 2a^2=d^2$
$\Rightarrow\ a=\frac{d}{\sqrt{2}}$
$\Rightarrow\ a=\frac{d\sqrt{2}}{2}\ \ $ by rationalizing the denominator.
So, length of the side as a function of the diagonal is, $\ \ a(d)=\frac{d\sqrt{2}}{2}$
$2.$ We know that the area of a square is equal to the square of its length of a side.
$A=a^2$
$\Rightarrow\ A=(\frac{d\sqrt{2}}{2})^2$
$\Rightarrow\ A=\frac{d^2\cdot 2}{4}$
$\Rightarrow\ A=\frac{d^2}{2}$
So, area as a function of diagonal is, $A(d)=\frac{d^2}{2}$
