Consider set A and function that are defined as follows: $A = \{ [,]^T \hspace{0.2cm}| \hspace{0.2cm}^2 + ^2 ≤ 1 \},\hspace{1cm} ([, ]^T) = [1, 2][,]^T + 1.$
Then what are the members of set B defined as follows:
$B= \{ ([, ]^T) \hspace{0.2cm}| \hspace{0.2cm}[,]^T \in A \}$
Your question is about finding the range of a linear function $$ f(x,y)=x+2y+1 $$
on the unit circle $A$, $x^2+y^2\le 1$. You can take the family of level curves (straight lines) $x+2y+1=k$ and find the values of $k$ for which the line touches the circle. Those will correspond to the max and min values of $f$. If you set up the system $x+2y+1=k$, $x^2+y^2=1$ you get a quadratic equation for $y$ $$ 5y^2-4(k-1)y+(k-1)^2-1=0 $$ There is only one intersection iff the discriminant is zero, that is $$ 16(k-1)^2-20[(k-1)^2-1]=0 $$
which gives $k=\pm\sqrt{5}+1$. So the desired range is $B=[-\sqrt{5}+1,\sqrt{5}+1]$.