I A Maron Solutions for Chapter: Introduction to Mathematical Analysis, Exercise 2: Function. Domain of Definition

In the square $A B C D$ with side $A B=2$ a straight line $MN$ is drawn perpendicularly to $A C$. Denoting the distance from the vertex $A$ to the line $MN$ as $x$, express through $x$ the area $S$ of the triangle $AMN$ cut off from the square by the straight line $M N$. Find this area at $x=\frac{\sqrt{2}}{2}$ and at $x=2$ 

Bring the number ${\alpha }_n$, which is equal to the $nth$ decimal place in the expansion of $\sqrt{2}$ into a decimal fraction, into correspondence with each natural number $n$. This gives us a certain function ${\alpha }_n=\phi \left(n\right)$. Calculate $\phi \left(1\right),\phi \left(2\right),\phi \left(3\right),\phi \left(4\right)$.

Calculate $f\left(x\right)=\frac{49}{x^2}+x^2$ at the points for which $\frac{7}{x}+x=3$.

Find a function of the form $f\left(x\right)=a{x}^2+bx+c$, if it is known that $f\left(0\right)=5 ; f(-1)=10 ; f\left(1\right)=6 .$

Find a function of the form $f\left(x\right)=a+b{c}^x(c>0)$  if$f\left(0\right)=15 ; f\left(2\right)=30 ; f\left(4\right)=90$

Find $\phi \left[\psi \right(x\left)\right]$and $\psi \left[\phi \right(x\left)\right]$ if $\phi \left(x\right)=x^2 \& \psi \left(x\right)=2^x.$

Given the function
$f\left(x\right)=\frac{5x^2+1}{2-x}$
Find $f\left(3x\right);f\left(x^3\right);3f\left(x\right);\left[f\right(x){]}^3$

Let $f\left(x\right)=\left\{\begin{array}{ccc}{3}^x& \mathrm{at} & -1<x<0,\\ 4& \mathrm{at} & 0\le x<1,\\ 3x-1& \mathrm{at} & 1\le x\le 3.\end{array}\right.$
Find $f\left(2\right), f\left(0\right), f(0.5), f(-0.5), f\left(3\right)$.