In the previous section, we saw modulus function and signum function. In this section, we will see greatest integer function.
G. Greatest integer function
• The greatest integer function is a real valued function f: R→R defined as:
$\mathbf\small{\rm{f(x)=\left\lfloor x\right\rfloor }}$
• Details can be written in 10 steps:
1. Given that, it is a real valued function. That means, all values obtained after processing, must be real values.
2. Whatever be the value of input x, the output will be $\mathbf\small{\rm{\left\lfloor x\right\rfloor }}$
This can be explained in 4 steps:
(i) $\mathbf\small{\rm{\left\lfloor x\right\rfloor }}$ is an integer.
♦ An integer is a positive or negative real number.
♦ But it cannot have fractional or decimal parts.
(ii) The integer obtained by $\mathbf\small{\rm{\left\lfloor x\right\rfloor }}$ must not be greater than the input x
(iii) The integer obtained by $\mathbf\small{\rm{\left\lfloor x\right\rfloor
}}$ can be equal to the
input x
(iv) The integer obtained by $\mathbf\small{\rm{\left\lfloor
x\right\rfloor }}$ must be the largest possible integer which is lesser
than the input x
• For each input x, we must examine each of the above four conditions.
3. Let us see some examples:
Example 1:
This can be written in 6 steps:
(i) Let the input x be 4.7
• So the output, which is f(x), will be $\mathbf\small{\rm{\left\lfloor 4.7\right\rfloor
}}$
(ii) Applying the first condition, $\mathbf\small{\rm{\left\lfloor 4.7\right\rfloor
}}$ will be an integer.
(iii) Applying the second condition, $\mathbf\small{\rm{\left\lfloor 4.7\right\rfloor
}}$ will not be greater than 4.7
♦ So all integers 5, 6, 7, . . . can be ruled out.
(iv) Applying the third condition, $\mathbf\small{\rm{\left\lfloor 4.7\right\rfloor
}}$ can be equal to 4.7
♦ But 4.7 is not an integer. So this condition is not applicable.
(v) Applying the fourth condition, $\mathbf\small{\rm{\left\lfloor 4.7\right\rfloor
}}$ will be equal to 4
(vi) So we can write:
When x = 4.7, f(x) = $\mathbf\small{\rm{\left\lfloor x\right\rfloor = \left\lfloor 4.7\right\rfloor
}}$ = 4
Example 2:
This can be written in 6 steps:
(i) Let the input x be 11.99
• So the output, which is f(x), will be $\mathbf\small{\rm{\left\lfloor 11.99\right\rfloor
}}$
(ii) Applying the first condition, $\mathbf\small{\rm{\left\lfloor 11.99\right\rfloor
}}$ will be an integer.
(iii) Applying the second condition, $\mathbf\small{\rm{\left\lfloor 11.99\right\rfloor
}}$ will not be greater than 11.99
♦ So all integers 12, 13, 14, . . . can be ruled out.
(iv) Applying the third condition, $\mathbf\small{\rm{\left\lfloor 11.99\right\rfloor
}}$ can be equal to 11.99
♦ But 11.99 is not an integer. So this condition is not applicable.
(v) Applying the fourth condition, $\mathbf\small{\rm{\left\lfloor 11.99\right\rfloor
}}$ will be equal to 11
(vi) So we can write:
When x = 11.99, f(x) = $\mathbf\small{\rm{\left\lfloor x\right\rfloor = \left\lfloor 11.99\right\rfloor
}}$ = 11
Example 3:
This can be written in 6 steps:
(i) Let the input x be -3.58
• So the output, which is f(x), will be $\mathbf\small{\rm{\left\lfloor -3.58\right\rfloor
}}$
(ii) Applying the first condition, $\mathbf\small{\rm{\left\lfloor -3.58\right\rfloor
}}$ will be an integer.
(iii) Applying the second condition, $\mathbf\small{\rm{\left\lfloor -3.58\right\rfloor
}}$ will not be greater than -3.58
♦ So all integers -3, -2, -1, 0, 1, 2, 3, 4, . . . can be ruled out.
(iv) Applying the third condition, $\mathbf\small{\rm{\left\lfloor -3.58\right\rfloor
}}$ can be equal to -3.58
♦ But -3.58 is not an integer. So this condition is not applicable.
(v) Applying the fourth condition, $\mathbf\small{\rm{\left\lfloor -3.58\right\rfloor
}}$ will be equal to -4
(vi) So we can write:
When x = -3.58, f(x) = $\mathbf\small{\rm{\left\lfloor x\right\rfloor = \left\lfloor -3.58\right\rfloor
}}$ = -4
Example 4:
This can be written in 5 steps:
(i) Let the input x be 8
• So the output, which is f(x), will be $\mathbf\small{\rm{\left\lfloor 8\right\rfloor
}}$
(ii) Applying the first condition, $\mathbf\small{\rm{\left\lfloor 8\right\rfloor
}}$ will be an integer.
(iii) Applying the second condition, $\mathbf\small{\rm{\left\lfloor 8\right\rfloor
}}$ will not be greater than 8
♦ So all integers 9, 10, 11, 12, . . . can be ruled out.
(iv) Applying the third condition, $\mathbf\small{\rm{\left\lfloor 8\right\rfloor
}}$ can be equal to 8
♦ 8 is indeed an integer. So this condition is applicable.
(v) So we can write:
When x = 8, f(x) = $\mathbf\small{\rm{\left\lfloor x\right\rfloor = \left\lfloor 8\right\rfloor
}}$ = 8
Example 5:
This can be written in 5 steps:
(i) Let the input x be -7
• So the output, which is f(x), will be $\mathbf\small{\rm{\left\lfloor -7\right\rfloor
}}$
(ii) Applying the first condition, $\mathbf\small{\rm{\left\lfloor -7\right\rfloor
}}$ will be an integer.
(iii) Applying the second condition, $\mathbf\small{\rm{\left\lfloor -7\right\rfloor
}}$ will not be greater than -7
♦ So all integers 9, 10, 11, 12, . . . can be ruled out.
(iv) Applying the third condition, $\mathbf\small{\rm{\left\lfloor -7\right\rfloor
}}$ can be equal to -7
♦ -7 is indeed an integer. So this condition is applicable.
(v) So we can write:
When x = -7, f(x) = $\mathbf\small{\rm{\left\lfloor x\right\rfloor = \left\lfloor -7\right\rfloor
}}$ = -7
4. Proceeding like this, we will get infinite number of ordered pairs. All
those ordered pairs should be included in the set f.
• So we can write: f = {. . . , (-7,-7), (-3.58, -4), (4.7, 4), (8, 8), (11.99, 11), . . .}
5. The above set f is written in roster form. But we have to remember an important point. It can be written in 3 steps:
(i) Both elements of the ordered pairs are real numbers.
• The second element y can take only integers (set Z). So we need not worry about y values.
(ii) Since x value can be any real number, there will be integers,
negative values, positive values, fractions, decimals, recurring
decimals, numbers like √2, √5, π etc.,. We cannot think of a definite sequence to write them.
(iii) So it is better to use set builder form to write f.
6. In the set builder form, we can write:
f = {(x,y) : x ∈ R, y ∈ Z}
• That means:
♦ The set f contains all ordered pairs (x,y) such that,
♦ x is a real number,
♦ y is an integer
7. Once we write the set f, we can write the domain and range of f.
(i) First we will write the domain:
• Domain of f is the set containing all the first elements of the ordered pairs in f.
• In our present case, there are infinite number of ordered pairs. So there will be infinite number of first elements.
•
We saw that all the first elements are real numbers. Since they are
real numbers, there will be integers,
negative values, positive values, fractions, decimals, recurring
decimals, numbers like √2, √5, π etc.,. We cannot think of a definite
sequence to write them. So it is better to use set builder form rather
than the roster form.
• We can write:
♦ Domain of f = {x : x ∈ R}
• That means:
♦ The domain of f will contain all real numbers.
(ii) Next we will write the range:
• Range of f is the set containing all the second elements of the ordered pairs in f.
• In our present case, there are infinite number of ordered pairs. So there will be infinite number of second elements.
•
We saw that the second elements can be only integers.
• We can write all integers in a definite pattern. So we can use the roster form.
• We can write:
♦ Range of f = {. . . , -3, -2, -1, 0, 1, 2, 3, 4, . . .}
• We can write it in the set builder form also:
♦ Range of f = {y: y ∈ Z}
• That means:
♦ The domain of f will contain all y such that, y is an integer.
8. We can make a table using the x and y values in the set f. Such a table is convenient to draw the graph of the function.
• Note that, to input for x, we choose convenient numbers from the set R.
• It is better not to choose numbers with recurring decimals. They will be difficult to plot.
 | |
| Table 2.9 |
9. The red lines in fig.2.17(a) below, is the graph of the function.
 |
| Fig.2.17 |
• We can write some peculiarities of these red lines. They can be written in 4 steps:
(i) The lines are small segments. They do not extend to infinity.
(ii) Consider the green vertical dashed line in fig.b. This line is drawn at x = 3.25
• That means, we are giving an input x value of 3.25
• When this x value is processed, we get the f(x) as 3
• Whatever real value (between 3 and 4) that we input for x, we will get f(x) as 3
♦ All y values in that red segment are indeed 3
♦ If the x value is not between 3 and 4, this red segment is not applicable.
• What if the x value is 4?
♦ Remember that, if x = 4, f(x) will be 4.
♦ This red segment do not have y = 4
♦ So if the x value is 4, this red segment is not applicable.
✰ We have to look at the next higher red segment.
♦ That is why a hollow circle is given at the right end of this red segment.
(iii) Consider the magenta vertical dashed line in fig.b. This line is drawn at x = -1.5
• That means, we are giving an input x value of -1.5
• When this x value is processed, we get the f(x) as -2
• Whatever real value (between -2 and -1) that we input for x, we will get f(x) as -2
♦ All y values in that red segment are indeed -2
♦ If the x value is not between -2 and -1, this red segment is not applicable.
• What if the x value is -1?
♦ Remember that, if x = -1, f(x) will be -1.
♦ This red segment do not have y = -1
♦ So if the x value is -1, this red segment is not applicable.
✰ We have to look at the next higher red segment.
♦ That is why a hollow circle is given at the right end of this red segment.
10. The greatest integer function is also known as step function because, it's graph looks like steps. This function has many applications in science and engineering.
In
the next
section, we will see algebra of real functions.
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