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Step Functions - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

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!

Some functions are not easily written as a formula. On a graph, a step function looks like a flight of stairs. The graphs of step functions have lines with an open circle on one end and a closed circle on the other to indicate inclusion, like number line inequality graphs. A rounding step function tells us to round a decimal number to the next whole integer or the previous whole integer.

We're now going to talk about step functions which are a different function than what we're used to. What we're used to is general curves that sort of flow so we've reached to a problem which is continuous, a cubic which is continuous, square root, everything is just sort of this fluid curve. And step functions are not the same, okay they're basically going to sort of look like a stair case if and when we graph them. The first one we want to look at is what's called the greatest integer function okay and it's just sort of weird bracket system where it sort of looks like an absolute value but you have another bracket in it. Just call the greatest integer function and what is, is basically saying give the greatest integer less and or equal to the number in it.
Okay so doing a couple of examples just so you can see how this works. This is saying the greatest integer less and or equal to 4.7 and if you remember integers are just any sort of whole number positive or negative. So what we're looking for is the greatest whole integer less than or equal to 4.7, 4.7 isn't an integer on its own right so we have to go to the next smallest one which is down to 4. Same logic by 12.1, 12.1 isn't an integer so we need to go down to the next largest integer which is going to be 12. 9, 9 is an integer, so the greatest integer less than or equal to 9 is obviously just going to be 9. And the one that always tricks students is negative 3.5, and if you think about how we're doing these on a number line, we have a number line say like 4.7 is over here we're going down to 4, 12.1 is over here we're going down to 12.
Negative 3.5 is going to be somewhere over here and we're not going to change our direction of smaller. Smaller is always down the number line to more negative numbers, so we go from negative 3.5 down and we end up at negative 4. Okay if you go to negative 3 you're actually making a number bigger because negative 3 isn't larger than negative 3.5. Even though numerically it's sort of weird just think about it for a second look at a number line and you'll see that okay negative 4 is actually smaller than negative 3.5. So really what a step function is at one example is the greatest integer function which is basically looking at a number line and picking the greatest integer less than or equal to whatever number you are given.

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