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Step Functions - Problem 1
University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
Graphing the greatest integer function. What we're going to do now is take a look at the greatest integer function and see what the graph looks like.
So what I'm going to do is basically start with a couple of points, talk about how we evaluate it and then go on and plot the graph. So let's just start at the origin. Greatest integer less than or equal to 0, 0 is an integer so the origin is going to be on the graph.
Going up numerically point 1, greatest integer less than or equal to point 1 goes back to 0, point 2 back to 0 so on and so forth all the way up to point 999, it still goes back down to zero, but the second we reach 1, our greatest integer function jumps up to 1, so actually what we end up with is a horizontal line on the x axis up and until 1 where I'm going to throw in a open circle which means that we don't include that point and then the graph jumps up to 1. Greatest integer of 1 point 1, back to 1, 1.2 back to 1, 1.99 back to 1 until we get to 2.
Once we reach 2, the graph is again going to have an open circle and jump up to 2. The same thing for all the 2s, so we're going to go down to 2, the second we reach 3 open circle up to 32 so on and so forth. Doing the same thing for our negative numbers. Negative .1, remember negative numbers are still going to go down to number line so negative .1 actually drops to -1 and what we end up with is just a continuation of the staircase going the other direction.
So what you have when you have the graph of a greatest integer function is basically a staircase. It's called the step function for a reason as it looks like the staircase and that's how we graph the greatest integer function.
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