Sketch a graph for each of the functions defined in 6—9 below.

7. h ( x ) = ⌈ x ⌉ − ⌊ x ⌋ for each real number x

Graph each of the function defined in 6−9 below.

h(x)=⌈x⌉−⌊x⌋ for all real numbers x.

Given information:

Consider the function.

For all real numbers

h(x)=⌈x⌉−⌊x⌋

The objective is to graph this function.

Concept used:

Calculation:

The objective is to graph this function.

Let x be a real number.

A floor function of x is denoted by f(x)=⌊x⌋ and is defined by the greatest integer that is less than or equal to x.

A ceiling function of x is denoted by f(x)=⌈x⌉ and is defined by the smallest integer that is greater than or equal to x.

The function h(x)=⌈x⌉−⌊x⌋ is the difference of ceiling function and floor function.

Now, examine the function h(x) for the small interval x∈(0,1).

⌈0.5⌉=1 and ⌊0.5⌋=0h(0.5)=⌈0.5⌉−⌊0.5⌋=1−0=1

Now, examine the function h(x) for the small interval x∈(−1,0).

⌈−0.5⌉=1 and ⌊−0.5⌋=−1h(−0.5)=⌈−0.5⌉−⌊−0.5⌋=0−(−1)=1

Next, examine the function h(x) for integer value.

x=−1

⌈−1⌉=−1 and ⌊−1⌋=−1h(−1)=⌈−1⌉−⌊−1⌋=−1−(−1)=0

Next, examine the function h(x) for integer value.

x=1⌈1⌉=1 and ⌊1⌋=1h(1)=⌈1⌉−⌊1⌋=1−(1)=0

Define a step function h(x) as below