I'm REALLY struggling to wrap my head around the leftmost and rightmost points of a function.

Question

I knew I had to ask for help when I was stuck for two and a half hours on one homework question that looked something like this:

f(x) = {-2x for x<1; 2x for >= 1

And I just could NOT figure it out. I looked at the provided examples, guided solutions, even the full explanation on how to solve the question; NOTHING CLICKED. I was so stressed because I've NEVER struggled this much at math before. I even asked my professor for help and his explanation didn't help me either (because it was the same as what the homework provided). I just don't get it. No matter how hard I try and solve it, it just doesn't make sense. The solution? I never ended up finding it myself. I rearranged all the numbers, I just kept thinking and thinking and thinking, and I can't figure out how they got to the solution.

Please, someone help me. I might honestly fail this class if I can't figure this out since we're not even halfway through the course and I already feel like I'm sinking into quicksand.

Answer 1

The function f(x) is a piecewise function. That means that it uses a different function description depending on what piece the value of x falls into.

For the function:

    f(x) = -2*x  for x<1

    f(x) = 2*x   for x>=1

When x < 1, we compute f(x) using -2*x. When x>=1, we compute f(x) using 2*x.

Note that the first expression is open-ended (that is, it does not include the endpoint). The second expression, however, includes the endpoint at x=1, so f(1) is computed using the second expression, not the first expression.

If both expressions are closed-ended and evaluate to the same value of f(1), we still have a function. If both functions are closed-ended and evaluate to a different value of f(1), THIS IS NOT A FUNCTION (there are two values of f(1) for one value of x).

Closed-ended line segments end with a point. Open-ended line segments are just a line.