Is algebraic geometry just linear algebra?
2 answers:
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The definition just says that the graph is not all the same function.
It's pieces of two different functions ... the function changes at the
line x = -1 . The graph looks something like the attached picture.
The function on the left side is y = x .
That's the graph starting from 10 miles left of the graph, and
all the way up to x = -1 . Surely you know how to draw y = x .
At x = -1, it changes to the other function.
The function on the right side is y = -x .
That's the graph starting from x = -1 and all the way out to
10 miles away on the right, or farther.
Surely you know how to draw y = -x .
That's all there is to it. Different functions on different parts of the plane.
And as you can see on the attached picture, there's no guarantee that
they have to meet at the same point. Sometimes they do and sometimes
they don't. It just depends on how they're defined.
Like, if the right-side function had been y = -x - 2 , then the two pieces
would meet, and the whole graph would look like a big upside-down
letter ' V ', with its vertex at the point (-1, -1) .
It's pieces of two different functions ... the function changes at the
line x = -1 . The graph looks something like the attached picture.
The function on the left side is y = x .
That's the graph starting from 10 miles left of the graph, and
all the way up to x = -1 . Surely you know how to draw y = x .
At x = -1, it changes to the other function.
The function on the right side is y = -x .
That's the graph starting from x = -1 and all the way out to
10 miles away on the right, or farther.
Surely you know how to draw y = -x .
That's all there is to it. Different functions on different parts of the plane.
And as you can see on the attached picture, there's no guarantee that
they have to meet at the same point. Sometimes they do and sometimes
they don't. It just depends on how they're defined.
Like, if the right-side function had been y = -x - 2 , then the two pieces
would meet, and the whole graph would look like a big upside-down
letter ' V ', with its vertex at the point (-1, -1) .