Step Functions

Vocabulary Step Functions- Functions whose graphs look like steps Greatest Integer Function- the function such that for every real number x, f(x) is the greatest integer less than or equal to x, also called rounding down or the floor function, denoted by ⌊x⌋ or INT(x) Discontinuous- Graph that cannot be drawn without lifting your pencil off the paper Points of Discontinuity- The values of x at which you lift your pencil Continuous- Graph has no points of discontinuity Ceiling Function- The function which pairs each number x with the smallest integer greater than or equal to x, also called rounding up, denoted by ⌈x⌉

Greatest Integer Function (Step Function)

Greatest Integer Function and Ceiling Function For ⌈x⌉ and ⌊x⌋ Domain: all real numbers Range: all Integers Range subject to change with multiplication/division/addition/subtraction ⌈x⌉=-⌊-x⌋

Greatest Integer Function INT(.9999)= ⌊-3.4⌋= INT(-2)= ⌊2.5⌋=

Using Calculator to find Greatest Integer Press MATH button Go Right one column to NUM Select option 5: int( Press ENTER Plug in x Press ENTER Get Answer

Applying the Greatest Integer Function A DVD costs $3 to rent for a night. How many DVDs can be rented if there is $35 in your wallet?

Other Greatest Integer Examples

Ceiling Functions ⌈2.1⌉= ⌈-4.4⌉= ⌈3⌉= ⌈-0.9⌉

Using Calculator to find Ceiling Function Follow all steps to find Greatest Integer Function Add 1 to the answer at the end Exception: if you are given an integer, than, like greatest integer, ceiling function does nothing to the number

Homework Examples F(x) = ⌈2x⌉ + ⌊x+1⌋ F(2) F(1.3) F(-0.5)

Applying the Ceiling Function A passenger train carries 200 people. How many passenger trains are required if there are 912 people looking to ride the train?

Phone Company A phone company charges 65 cents a minute for all calls from the US to Germany. They round all calls up to the nearest 5 seconds. Write an equation to give the cost C of calls that last t seconds.

Graphing a Step Function Adding, inside or outside the step function, will not change the step length or the distance between the steps, but it will move the graph up. Subtracting, inside or outside the step function, will not change the step length or the distance between the steps, but it will move the graph down.

Identifying Points of Discontinuity A point of discontinuity is a point on the graph that if you were tracing the graph with your pencil, you would have to pick your pencil up and move to another spot on the paper to continue tracing. When listing points of discontinuity, always use the x value of where you have to pick the pencil up

Finding Domain and Range Domain is all real numbers Range is found by finding a pattern for all the y values that are covered on the graph

Examples

Homework Worksheet 2-7