question about greatest integer function ~~~~~~~?

Question

it says to graph the equation..
nd they give you lyk

g(x)=[[3x]]+4
it says u can use calc
but teacher said she wants us to do it by hand
and i have no clue ><
any ideas?

Answer 1

Hi,

The constant on the end, 4, is the y intercept. If you take the expression inside the greatest integer function symbol and set it equal to one and solve, you get 3x = 1 so x = ⅓. This means each "step" will be ⅓ of a block long.

Your first step starts at (0,4) and go to a hollow dot at (⅓,4). Then the graph jumps to (⅓,5) and has another horizontal step to a hollow dot at (⅔,5). then the graph jumps to (⅔,6) and starts the next step, ⅓ of a block long to a hollow dot at (1,6).

This pattern will continue.

I hope that helps!! :-)

Answer 2

property of greatest integer function [[f(x)]]

it has a value of d .... whenever d-1 < f(x) ≤ d

thus consider first [[3x]]

it is ...
-2 ...... -3 < 3x ≤ -2
-1 ...... -2 < 3x ≤ -1
0 ...... -1 < 3x ≤ 0
1 ....... 0 < 3x ≤ 1
2 ....... 1 < 3x ≤ 2
3 ....... 2 < 3x ≤ 3
... and so on....


thus
[[ 3x ]] = ..
-2 ...... -1 < x ≤ -2/3
-1 ...... -2/3 < x ≤ -1/3
0 ...... -1/3 < x ≤ 0
1 ....... 0 < x ≤ 1/3
2 ....... 1/3 < x ≤ 2/3
3 ....... 2/3 < x ≤ 1


finally...
g(x) = [[3x]] + 4 =
......
2 ...... -1 < x ≤ -2/3
3 ...... -2/3 < x ≤ -1/3
4 ...... -1/3 < x ≤ 0
5 ....... 0 < x ≤ 1/3
6 ....... 1/3 < x ≤ 2/3
7 ....... 2/3 < x ≤ 1
... and so on... notice that the length of the interval is constant and the included endpoint is always on the right... thus you can extend the graph to the left and to the right...


§

Answer 3

The easiest thing to do is to make a table of values. So:

x y
0 4
1 7
2 10
-1 1
-2 -2

If you were to graph those points, it should become readily apparent what the pattern is to the steps. Although, if I were you I would plug some fractions in just to be sure that the steps don't end in the middle, but by the looks of the problem I don't think it does.

Any other question, feel free to e-mail me
[e-mail]
or AIM me
nobody0045

Answer 4

Make your own graph.

Plug in numbers for x and plot them on the graph, like if x=0, g(x) = 4, if x = 2, g(x) = 10, if x = -1, g(x) = 7.

It should be shaped like a capital V and the vertex should be at g(x) = 4.

Answer 5

graphing f(x) = [x], you know you get little horizontal bars, 1 unit long, solid dot at left end, hollow dot at right, stepping up like stairs at integer values of x: bar from (0,0) to (1,0), from (1,1) to (2,1), and so on.

adding 4 just moves them all up 4 units.

the multiplication by 3 compresses everything horizontally. without the +4 part, h(x) = [3x] has bars from (0,0) to (1/3, 0), from (1/3, 1) to (2/3, 1), from (2/3, 2) to (1/,2), and so on. Adding 4 just moves it all up 4 units.