In the previous section, we saw rational function. In this section, we will see Modulus function and Signum function.
E. Modulus function
• A modulus function is a real valued function f: R→R defined by:
$\mathbf\small{\rm{f(x)=|x|}}$
• Details can be written in 12 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 x, the y value will be the modulus of that x value.
• This can be explained in two steps:
(i) If we take a positive number for input x, we must first write the modulus of that number. The y value will be this modulus.
♦ For example, let us take '8' for input x. The modulus of 8 = |8| = 8
♦ So f(x) = 8
(ii) If we take a negative number for input x, then also we must first write the modulus of that number. The y value will be this modulus.
♦ For example, let us take '-7' for input x. The modulus of -7 = |-7| = 7
♦ So f(x) = 7
3. The above information can be written in algebraic form as:
$f(x) =
\begin{cases}
x, & \text{if}\; x \geq 0 \\
-x, & \text{if} \; x\lt 0
\end{cases}$
• We see two conditions on the right side of the curly bracket.
♦ The first condition is same as what we wrote in 2(i)
♦ The second condition is same as what we wrote in 2(ii)
• The second condition needs special attention. It has a '-' sign in front of 'x'. It will become clear if we consider an example:
♦ Let us take input x as -7
♦ Then f(x) = -x = -(x) = -(-7) = 7
♦ So the '-' sign is important.
4. The function can be defined by writing the ordered pairs which satisfy that function.
• So our next task is to find those ordered pairs.
5. It is given that, f : R→R
• This indicates that,
♦ the first elements of the ordered pairs (input x values) should be taken from the set R.
♦ the second elements (resulting y values) should be present in the set R
6.
The set R is the set of real numbers. It will include integers,
negative values, positive values, fractions, decimals, recurring
decimals, numbers like √2, √5, π etc.,. In short, R will contain every
value which can be plotted on a number line. Recall that we plotted √2,
√5, π etc., in our previous classes.
• Since different types of numbers are present in R, we will choose some convenient numbers at random.
• Let x = -7
♦ This x is processed as follows:
♦ Here x < 0. So we use the second condition
♦ f(x) = -x = -(x) = -(-7) = 7
♦ f(x) = f(-7) is the 'y value' when 'x value' is -7
♦ So we get an ordered pair (x,y) as: (-7, 7)
• Let x = -5
♦ This x is processed as follows:
♦ Here x < 0. So we use the second condition
♦ f(x) = -x = -(x) = -(-5) = 5
♦ f(x) = f(-5) is the 'y value' when 'x value' is -5
♦ So we get another ordered pair (x,y) as: (-5, 5)
• Let x = 0
♦ This x is processed as follows:
♦ Here x ≥ 0. So we use the first condition
♦ f(x) = x = 0
♦ f(x) = f(0) is the 'y value' when 'x value' is 0
♦ So we get another ordered pair (x,y) as: (0, 0)
• Let x = 2
♦ This x is processed as follows:
♦ Here x ≥ 0. So we use the first condition
♦ f(x) = x = 2
♦ f(x) = f(2) is the 'y value' when 'x value' is 2
♦ So we get another ordered pair (x,y) as: (2, 2)
• Let x = 5
♦ This x is processed as follows:
♦ Here x ≥ 0. So we use the first condition
♦ f(x) = x = 5
♦ f(x) = f(5) is the 'y value' when 'x value' is 5
♦ So we get another ordered pair (x,y) as: (5, 5)
• We see that, whatever be the value of x, the value of y will be the modulus of that x value.
7. 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), (-5, 5), (0, 0), (2, 2), (5, 5), . . .}
8. 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.
(ii) 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.
(iii) So it is better to use set builder form to write f.
9. In the set builder form, we can write:
f = {(x,y) : x ∈ R, y = |x|, |x| ∈ R}
• That means:
♦ The set f contains all ordered pairs (x,y) such that,
♦ x is a real number,
♦ y is the modulus of x,
♦ modulus of x is a real number
10. 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 all the first elements are real numbers. Since the second
elements are moduli, there will not be any negative numbers. But there
will be integers, fractions, decimals, recurring
decimals 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:
♦ Range of f = {x : x ∈ R+}
• That means:
♦ The range of f will contain all positive real numbers.
11. 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 2.7 |
12. The red lines in fig.2.15(a) below, is the graph of the function.
 |
| Fig.2.15 |
• The reader is advised to prepare a table and draw the graph in his/her own graph books.
• We can write some peculiarities of these red lines. They can be written in 4 steps:
(i) The red lines of f(x) = |x| always pass through the origin (0,0)
(ii) Mark any point on the red lines. Note the coordinates of that point.
♦ The y coordinate will be the modulus of the x coordinate.
♦ This is shown in fig.b
♦ The green vertical dashed line shows 8
✰ The green horizontal dashed line shows the modulus 8
♦ The magenta vertical dashed line shows -6
✰ The magenta horizontal dashed line shows the modulus 6
(iii) The red lines never go below the x axis. This is because, there will never be any negative y values.
(iv) We see arrows at both ends of the red lines.
• The left arrow indicates that:
The line can extend up to a point where x = -∞ and y = modulus of that x value.
• The right arrow indicates that:
The line can extend up to a point where x = +∞ and y = modulus of that x value.
F. Signum function
• A signum function is a real valued function f: R→R defined by three conditions:
$f(x) =
\begin{cases}
1, & \text{if}\; x > 0 \\
0, & \text{if} \; x = 0 \\
-1, & \text{if} \; x < 0
\end{cases}$
• Details can be written in 12 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 x, we must examine each of the three conditions.
• This can be explained in three steps:
(i) If we take a positive number for input x, the y value will be 1
♦ For example, let us take '8' for input x. Then y = f(x) = 1
(ii) If we take zero for input x, the y value will be 0
(iii) If we take a negative number for input x, the y value will be -1
♦ For example, let us take '-6' for input x. Then y = f(x) = -1
3. The function can be defined by writing the ordered pairs which satisfy that function.
• So our next task is to find those ordered pairs.
4. It is given that, f : R→R
• This indicates that,
♦ the first elements of the ordered pairs (input x values) should be taken from the set R.
♦ the second elements (resulting y values) should be present in the set R
5.
The set R is the set of real numbers. It will include integers,
negative values, positive values, fractions, decimals, recurring
decimals, numbers like √2, √5, π etc.,. In short, R will contain every
value which can be plotted on a number line. Recall that we plotted √2,
√5, π etc., in our previous classes.
• Since different types of numbers are present in R, we will choose some convenient numbers at random.
• Let x = -7
♦ This x is processed as follows:
♦ Here x < 0. So we use the third condition
♦ f(x) = -1
♦ f(x) = f(-7) is the 'y value' when 'x value' is -7
♦ So we get an ordered pair (x,y) as: (-7, -1)
• Let x = -5
♦ This x is processed as follows:
♦ Here x < 0. So we use the third condition
♦ f(x) = -1
♦ f(x) = f(-5) is the 'y value' when 'x value' is -5
♦ So we get another ordered pair (x,y) as: (-5, -1)
• Let x = 0
♦ This x is processed as follows:
♦ Here x = 0. So we use the second condition.
♦ f(x) = x = 0
♦ f(x) = f(0) is the 'y value' when 'x value' is 0
♦ So we get another ordered pair (x,y) as: (0, 0)
• Let x = 2
♦ This x is processed as follows:
♦ Here x > 0. So we use the first condition
♦ f(x) = 1
♦ f(x) = f(2) is the 'y value' when 'x value' is 2
♦ So we get another ordered pair (x,y) as: (2, 1)
• Let x = 5
♦ This x is processed as follows:
♦ Here x > 0. So we use the first condition
♦ f(x) = 1
♦ f(x) = f(5) is the 'y value' when 'x value' is 5
♦ So we get another ordered pair (x,y) as: (5, 1)
• We see that,
♦ If the input x is a positive value, y will be +1
♦ If the input x is zero, y will be zero
♦ If the input x is a negative value, y will be -1
6. 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,-1), (-5, -1), (0, 0), (2, 1), (5, 1), . . .}
7. 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 three values 1, 0 and -1. 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.
8. In the set builder form, we can write:
f = {(x,y) : x ∈ R, y = +1 if x > 0, y = 0 if x = 0, y = -1 if x < 0}
• That means:
♦ The set f contains all ordered pairs (x,y) such that,
♦ x is a real number,
♦ y can be +1, 0 or -1
9. 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 +1, 0 or -1.
♦ There will be infinite number of +1
♦ There will be only one 0
♦ There will be infinite number of +1
• But when we write a set, repeating elements are written only once. So we can easily write in roster form.
• We can write:
♦ Range of f = {1, 0, -1}
10. 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 2.8 |
11. The red lines in fig.2.16(a) below, is the graph of the function.
 |
| Fig.2.16 |
• The reader is advised to prepare a table and draw the graph in his/her own graph books.
• We can write some peculiarities of these red lines. They can be written in 4 steps:
(i) The top red line is obtained when we use positive x values for input.
• This line never goes to the left side of the y axis because, no negative x values are used for this line.
• In fact, this line do not even touch the y axis. Because, (x = 0) is not used for this line. Remember that this line is for x > 0. That means, only those values which are greater than zero can be used for this line.
• Since it cannot touch the y axis, a hollow red circle is given at the left end of this line. The hollow red circle indicates that, the point inside the circle is not included in the graph.
(ii) The red filled circle at (0, 0) is obtained when we use (x = 0) for input.
• Even if it is a single point, it is a graph. It is the graph satisfying the second condition.
• We can write:
The graph satisfying the second condition contains only one point.
(iii) The bottom red line is obtained when we use negative x values for input.
• This line never goes to the right side of the y axis because, no positive x values are used for this line.
•
In fact, this line do not even touch the y axis. Because, (x = 0) is
not used for this line. Remember that this line is for x < 0. That
means, only those values which are lesser than zero can be used for
this line.
• Since it cannot touch the y axis, a hollow red circle is
given at the right end of this line. The hollow red circle indicates
that, the point inside the circle is not included in the graph.
(iv) Mark any point on the red lines. Note the coordinates of that point.
♦ The y coordinate will be either -1 or +1
♦ This is shown in fig.b
♦ The green vertical dashed line shows 1.5
✰ This line meets the red line at a point where y coordinate is +1
♦ The magenta vertical dashed line shows -1.25
✰ This line meets the red line at a point where the y coordinate is -1
(v) We see one arrow each for the top and bottom red lines.
• The arrow of the top red line indicates that:
The line can extend up to a point where x = +∞ and y = +1
• The arrow of the bottom red line indicates that:
The line can extend up to a point where x = -∞ and y = -1
12. The signum function has many applications in science and engineering.
• If the output of a mathematical calculation process is +1, a machine can be instructed to start operation.
• If the output of a mathematical calculation process is -1, the machine can be instructed to stop operation.
• Let us see an example. It can be written in steps:
(i) Consider the function: $\mathbf\small{\rm{f(x)=\frac{x}{|x|}}}$
(ii) Let input x be -4
• Then $\mathbf\small{\rm{f(-4)=\frac{-4}{|-4|}=\frac{-4}{4}}}$ = -1
♦ The machine will stop operation.
(iii) Let input x be +7
• Then $\mathbf\small{\rm{f(+7)=\frac{+7}{|+7|}=\frac{+7}{7}}}$ = +1
♦ The machine will start operation.
In
the next
section, we will see Greatest integer function.
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