Step Functions Word Problem

Question

Johnny is very confused on why you have open ended sides on the left sometimes and then sometimes on the right. Explain to him in complete sentences how this situation occurs, and how Johnny might be able to straighten out the two cases.

Answer 1

There are also two different notations for step functions that tell you whether to round up or down, and thus whether the open circles belong on the left or right on the graph.

The ceiling function, denoted by ⌈x⌉ (notice it looks like a ceiling), is defined as the least integer greater than or equal to x, which means round up. The open holes on the steps of the graph would be on the left.

Contrarily, the floor function, denoted by ⌊x⌋ (notice it looks like a floor), is defined as the greatest integer less than or equal to x, which means round down. The open holes on the graph are now on the right.

Answer 2

With a step function of general form y = [x], for every value of x the value of y is equal to the greatest integer that is less than or equal to x. In this case, we would have open circles (depicting that the function exists for all values approaching that point but not at that specific point) on the right because that would be when we approach the next integer, ex: there would be an open circle at (5,4) for the general function above because as the function approaches x = 5, ex: x = 4.5, 4.7, 4.9, 4.999, the greatest integer less than or equal to the x value is still 4, so y will be 4. However the second the value of x reaches 5, the greatest integer less than or equal to x is 5, so the value of y would be 5, leaving an open circle at (5,4) because that value does not lie on the function (you can't input 5 for x and get 4 for y, it would be impossible).

We can then see the place of this open circle changing if we apply transformations to the general form of this function. For example, y = [-x] would give a mirror transformation of a reflection across the y-axis now putting the circles on the left. This is because we are now saying for all values of x, y is equal to the greatest integer that is less than or equal to the opposite sign value of x. For example, as x approaches 5 on this function coming from the right side this time (x decreasing) we would see that for x = 5.5, 5.3, 5.1, 5.00002, y is the greatest integer less than or equal to the opposite of those numbers which is -6 (for example the greatest integer less than or equal to -5.5 is -6), however when the value of x reaches 5, the greatest integer less than or equal to -5 is -5, so the y value at x = 5 will be -5, which would require a open circle at (5,-6) because the function does not exist at this point.

The best way to approach this when graphing a step function is to make a table or plug in some points for x and y values. You can then connect these points and get a general shape of the graph. Try to go in between integers to get some decimal values of x to plug in so you can see how the fraction values of x yield integer values of y. Also make sure to plug in all the integer values so that you can see what happens to the endpoints. Here is an example

y = [-2x]

X	Y
-2	4
-1.5	3
-1.25	2
-1	2
-0.5	1
0	0
0.5	-1
1	-2
1.25	-3
1.5	-3
2	-4

We can then find the x values of the endpoints which will be at every 0.5x. We can then look at our table and plot them and fill in a solid circle because we know the function exists at that point. Then determine which direction the line goes based off of points in between. In this case for example at x = 1.5, (1.5,-3), we put a solid circle and see that the line goes left from that point because moving left to x value 1.25, we see that the x value is the same as 1.5, so we can draw the line left. Then we must stop at x = 1 because we see that x does not equal -3 any more and must put an open circle there.

I hope this helps. If you have any other questions, let me know.

Answer 3

(5,7] is "open ended" in the sense of not including 5, just allowing numbers that approach 5 but do not include 5. [5,7) in contrast includes 5 and every number between 5 and 7, but does not include 7.

Jack is more than 5 years old, but no more than 7 years old . 5<J<7. 5 is less than Jack's age, but Jack's age is equal to or less than 7 years old.

Parentheses do not include the end point, while brackets do include the end point.

It's like a ledge, a cliff. If you are 5 feet closer, you fall off, but if you are more than 5 feet away, you don't fall off. Unless, you are 7 feet away from the cliff you are safe, but slightly more than 7 feet away, and you fall into another cliff on the other side of you. You're only safe if you are a distance D such that 5<D<7. D = 5 or less and you fall over the cliff. D= 7 you're safe, but D>7 you fall off. 5<D<7 is the same as D(5,7]

Or maybe consider a checking account that charges you fees if you have too little money in your account, and also fees if you have too much in your account (or related too little activity). Like the IRS, if you don't withhold enough from your paychecks you get a penalty fee, or if you get too much of a refund, they charge you with a fee. You're between a rock and a hard place. A limited interval, which could get you a fee depending on whether you go past the limit, or reach the limit. Past or reach the limit is like the bracket, parentheses, open ended or closed ended limits.