Why $S_1$ and $S_4$ are represented by lines in the region $R$?

Question

Here is the solution I am asking about:

Why $S_1$ and $S_4$ are represented by lines in the region $R$? I do not see how their equations are the equations of a line. Could someone explain this to me please?

Answer 1

As you know, an equation in the form $y = mx + b$ is the equation of a line. (Vertical lines cannot be put into this form, but for the moment that is not an issue since the lines we are looking for are not vertical.)

One line of this form is obtained by setting $m = b = 0.$ The equation of this line is simply $$ y = 0. $$

That line also happens to be the $x$-axis.

But the set of points satisfying this equation is an entire unbounded line, not a line segment. In order to describe a line segment, we need to put some bounds on the coordinates. For example, if we say $$ 0 \leq x \leq 1 $$ (which is actually a pair of inequalities, $0 \leq x$ and $x \leq 1$), then all the parts of the line $y = 0$ where $x < 0$ and all the parts where $x > 1$ are excluded, leaving only the line segment consisting of the points $(0,0)$, $(1,0),$ and the points between those two points.

The problem solution you have shared gives the following description of the image of $S_1$:

I have added red boxes to point out the specific equation and inequalities that describe the image of $S_1$ and tell us that the image is a line segment. In fact these are exactly the equation and inequalities I used to describe this same segment earlier.

The description of the image of $S_4$ is near the end of the solution:

Again, I have put red boxes around the equation and inequalities that describe the line segment. Again we have $y = 0$ (the line segment is on the $x$-axis) but this time the inequalities are $-1 \leq x$ and $x \leq 0,$ which tell us that the line segment runs from $(-1,0)$ to $(0,0).$

We also have equations such as $x = u^2$ in the derivation of the image of $S_1,$ but all this does is to determine that $x$ cannot be negative. We need the additional information that $0 \leq u \leq 1$ to determine that $0 \leq x \leq 1,$ which gives us the actual endpoints of the line segment. Similarly, for the image of $S_4$ we have $x = -v^2,$ but we need to use the fact that $0 \leq v \leq 1$ in order to find that $-1 \leq x \leq 0,$ giving us the actual endpoints of that line segment. In both cases it is the equation $y = 0$ that tells us we have a line or part of a line.