14-­‐12-­‐11 The simplest greatest integer function, GIF for short is….
f(x) = [x]
World 3-1
The Greatest Integer Function: GIF
output
input
Square brackets [ ] always mean round down to
the nearest integer. THINK FLOOR
Eg 1. f(x) = [x]
f(3) = [3]
=3
Eg 2. Evaluate f(x)= [x]
f(4) = [4.4] f(-6) = [-6]
=4
= -6
-3
-2
0
2
1
g(x) = 2[x] +1
g(1.2) = 2[1.2] +1
= 2(1) + 1
=3
u(x) = 8[0.4x] - 2
u(5) = 8[0.4(5.1)] -2
3
Imagine a number line. The GIF always shift the value
left, to the nearest integer.
s(x) = 5[x-0.3] - 8
s(3.2) = 5[3.2-0.3] -8
= 5[2.9] - 8
= 5(2) - 8
=2
w(x) = -3[-4x] - 27
w(2.1) = -3[(-4)(2.1)] - 27
= 8[2.04] -2
=
=
=
=
= 8(2) -2
= 14
Eg. 4 Given the output, f(x), the input, x, can be solved
If [x] = i, and i is an integer, then
[-2x] = 4
2[x-4] = 10
2[x-4] < 10
2
2
x < -4
[x-4] ≥ 5
[x-4] < 5 +1
{x ∈ R | −5 ≤ x < −4}
x–4≥5
+4+4
x-4 < 6
+4 +4
x≥9
x < 10
x ≥ -5
x + 3 < -2 + 1
-3[-8.4] - 27
-3(-9) - 27
27 - 27
0
Eg. 5 Solve for x
2[x-4] ≥ 10
2
2
€ x + 3 ≥ -2
{x ∈ R | 4 ≤ x < 5}
€
{x ∈ R | i ≤ x < i +1}
[x+3] = -2
[x] = 4
x≥4 x<4+1
x<5
f(3.99) = [3.99]
=3
Eg. 3 Evaluate the GIF’s with transformations
f(-5.4) = [-5.8] f(-7.1) = [-7.1]
= -6
= -8
-1
f(3.5) = [3.5]
=3
{x ∈ R | 9 ≤ x <10}
[-2x] ≥ 4
[-2x] < 4+1
-2x ≥ 4
-2 -2
-2x < 5
-2
-2
x ≤ -2
x > -2
{x ∈ R | −2.5 < x ≤ −2}
Math Gods say flip sign
when dividing by a negative
€
1 14-­‐12-­‐11 Homework
Activity 1
Pg. 117 #1-5
Pg. 116 Act 1 #1
2