In the previous section, we saw constant function and polynomial function. In this section, we will see rational function.
D. Rational function
• A rational function is a real valued function f: R→R defined by:
$\mathbf\small{\rm{f(x)=\frac{p(x)}{q(x)}}}$
• Details can be written in 11 steps:
1. Given that, it is a real valued function. That means, all values obtained after processing, must be real values.
2. Let us examine the numerator and denominator:
• The numerator is p(x). It must be a polynomial function.
• The denominator is q(x). It must also be a polynomial function.
♦ This q(x) should not be equal to zero.
♦ If it is zero, the function f(x) cannot be defined.
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.
• We will take a simple rational function: f(x) = $\mathbf\small{\rm{\frac{1}{x}}}$
♦ Recall that all constants are polynomials. So '1' is a polynomial.
♦ For example, 4 can be written as: 4x0, which is (4 × 1) = 4
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:
♦ f(-7) = $\mathbf\small{\rm{\frac{1}{(-7)}}}$ = -0.143
♦ f(-7) is the 'y value' when 'x value' is -7
♦ So we get an ordered pair (x,y) as: (-7, -0.143)
• Let x = -5
♦ This x is processed as follows:
♦ f(-5) = $\mathbf\small{\rm{\frac{1}{(-5)}}}$ = -0.2
♦ f(-5) is the 'y value' when 'x value' is -5
♦ So we get an ordered pair (x,y) as: (-5, -0.2)
• Let x = 1
♦ This x is processed as follows:
♦ f(1) = $\mathbf\small{\rm{\frac{1}{(1)}}}$ = 1
♦ f(1) is the 'y value' when 'x value' is 1
♦ So we get an ordered pair (x,y) as: (1, 1)
• Let x = 2
♦ This x is processed as follows:
♦ f(2) = $\mathbf\small{\rm{\frac{1}{(2)}}}$ = 0.5
♦ f(2) is the 'y value' when 'x value' is 2
♦ So we get an ordered pair (x,y) as: (2, 0.5)
• Let x = 5
♦ This x is processed as follows:
♦ f(5) = $\mathbf\small{\rm{\frac{1}{(5)}}}$ = 0.2
♦ f(5) is the 'y value' when 'x value' is 0.2
♦ So we get an ordered pair (x,y) as: (5, 0.2)
• Let x = 10
♦ This x is processed as follows:
♦ f(10) = $\mathbf\small{\rm{\frac{1}{(10)}}}$ = 0.1
♦ f(10) is the 'y value' when 'x value' is 10
♦ So we get an ordered pair (x,y) as: (10, 0.1)
• We see that, whatever be the value of x, the value of y will be the reciprocal of that x value.
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,-0.143), (-5, -0.2), (1, 1), (2, 0.5), (5, 0.2), (10, 0.1), . . .}
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.
(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.
8. In the set builder form, we can write:
f = {(x,y) : x ∈ R, y = $\mathbf\small{\rm{\frac{1}{x}}}$, $\mathbf\small{\rm{\frac{1}{x}}}$ ∈ R}
• That means:
♦ The set f contains all ordered pairs (x,y) such that,
♦ x is a real number,
♦ y is the reciprocal of x,
♦ reciprocal of x is a real number
• But there is a problem. We cannot put x = 0. The function is not defined at x = 0.
• This problem can be solved in 3 steps:
(i) Consider the set R - {0}.
♦ Here we are subtracting the 'set containing zero' from the set R
(ii) The resulting set will not contain zero.
♦ We must take x only from R - {0}
(iii) So we must modify the above set builder form. We get:
f = {(x,y) : x ∈ R - {0}, y = $\mathbf\small{\rm{\frac{1}{x}}}$, $\mathbf\small{\rm{\frac{1}{x}}}$ ∈ R}
• That means:
♦ The set f contains all ordered pairs (x,y) such that,
♦ x is an element of R - {0},
♦ y is the reciprocal of x,
♦ reciprocal of x is a real number
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 (except zero). 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 - {0}}
• That means:
♦ The domain of f will contain all real numbers except zero.
(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 (except zero). Since the second
elements are reciprocals, they will also be real numbers. So there
will be integers, negative values, positive values, 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 = {$\mathbf\small{\rm{\frac{1}{x}}}$ : x ∈ R - {0}}
• That means:
♦ The range of f will contain reciprocals of all real numbers except zero.
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 - {0}.
• It is better not to choose numbers with recurring decimals. They will be difficult to plot.
 |
| Table 2.6 |
11. The red curves in fig.2.14(a) below, is the graph of the function.
 |
| Fig.2.14 |
• The graph in fig.b is drawn to a different scale. This can be explained in 5 steps:
(i) The y value is the reciprocal of x value. So we will need many small values on the y axis.
(ii) To achieve this within limited space, we change the scale of the y axis.
(iii) In the fig.2.14(a), we have:
♦ Scale of x axis as: 1 cm = 1 unit
♦ Scale of y axis as: 1 cm = 1 unit
(iv) In the fig.2.13(b), we have:
♦ Scale of x axis as: 1 cm = 1 unit
♦ Scale of y axis as: 1 cm = 0.25 unit
(v) So the graph in fig.b is a bit distorted. But the coordinates do not change.
• The reader is advised to prepare a table and draw the graph in his/her own graph books.
• We can write some peculiarities of this red curve. They can be written in 3 steps:
(i) The red curve of f(x) = $\mathbf\small{\rm{\frac{1}{x}}}$ never passes through the origin (0,0)
(ii) Mark any point on the red curve. Note the coordinates of that point.
♦ The y coordinate will be the reciprocal of the x coordinate.
♦ This is shown in fig.b
♦ The green vertical dashed line shows 2
✰ The green horizontal dashed line shows the reciprocal 0.5
♦ The magenta vertical dashed line shows -5
✰ The magenta horizontal dashed line shows the reciprocal -0.2
(iii) We see arrows at both ends of the red curves. There are four arrows. We will call them I, II, III and IV.
I. The top arrow of the right side red curve:
• This arrow indicates that, the curve can go very high up.
• This can be explained as follows:
♦ As we move along this curve from right towards the left, x takes values closer and closer to zero.
✰ For example, 0.00001 is closer to zero than 0.0001
♦ Values closer to zero are very small.
✰ The reciprocals of such small values will be very large.
♦ The reciprocals are y values.
✰ That means y values will be very large.
✰ They are very high up in the graph.
✰ This arrow indeed indicates that the graph will go very high up (y = +∞).
♦ As the x takes values closer and closer to zero, we get the feeling that, this end of the curve is going to touch the y axis.
✰ It will touch the y axis when x = 0
✰ However, x cannot become equal to zero.
✰ If it is zero, then the function can not be defined.
✰ This is because, $\mathbf\small{\rm{\frac{1}{x}}}$ becomes $\mathbf\small{\rm{\frac{1}{0}}}$
♦ So this end will never touch the y axis
✰ When we draw this graph on a sheet of paper or the computer screen, it appears to touch.
✰ But if we zoom in, we will see that, there is a little gap.
II. The bottom arrow of the right side red curve:
• This arrow indicates that, the curve can reach to the point where (x = +∞).
• This can be explained as follows:
♦ As we move along this curve from left towards the right, x take values closer and closer to +∞.
♦ Values closer to +∞ are very large.
✰ The reciprocals of such small values will be very small.
♦ The reciprocals are y values.
✰ That means y values will be very small.
✰ They get closer and closer to zero.
♦ As the y becomes closer and closer to zero, we get the feeling
that, this end of the curve is going to touch the x axis.
✰ It will touch the x axis when x = +∞
✰ However, x cannot become equal to +∞.
♦ So this end will never touch the x axis
✰ When we draw this graph on a sheet of paper or the computer screen, it appears to touch.
✰ But if we zoom in, we will see that, there is a little gap.
III. The top arrow of the left side red curve:
• This arrow indicates that, the curve can reach to the point where (x = -∞).
• This can be explained as follows:
♦ As we move along this curve from right towards the left, x take values closer and closer to -∞.
♦ Values closer to -∞ are very large (numerically).
✰ The reciprocals of such small values will be very small.
♦ The reciprocals are negative y values.
✰ That means negative y values will be very small.
✰ They get closer and closer to zero.
♦ As the y becomes closer and closer to zero, we get the feeling
that, this end of the curve is going to touch the x axis.
✰ It will touch the x axis when x = -∞
✰ However, x cannot become equal to -∞.
♦ So this end will never touch the x axis
✰ When we draw this graph on a sheet of paper or the computer screen, it appears to touch.
✰ But if we zoom in, we will see that, there is a little gap.
IV. The bottom arrow of the left side red curve:
• This arrow indicates that, the curve can go very low bottom.
• This can be explained as follows:
♦ As we move along this curve from left towards the right, x takes values closer and closer to zero.
✰ For example, -0.0001 is closer to zero than -0.00001
♦ Values closer to zero are very small.
✰ The reciprocals of such small values will be very large.
♦ The reciprocals are negative y values.
✰ That means negative y values will be very large (numerically).
✰ They are very low down in the graph.
✰ This arrow indeed indicates that the graph will go very low bottom (y = -∞).
♦ As the x takes values closer and closer to zero, we get the feeling
that, this end of the curve is going to touch the y axis.
✰ It will touch the y axis when x = 0
✰ However, x cannot become equal to zero.
✰ If it is zero, then the function can not be defined.
✰ This is because, $\mathbf\small{\rm{\frac{1}{x}}}$ becomes $\mathbf\small{\rm{\frac{1}{0}}}$
♦ So this end will never touch the y axis
✰ When we draw this graph on a sheet of paper or the computer screen, it appears to touch.
✰ But if we zoom in, we will see that, there is a little gap.
10. The rational function f(x) = $\mathbf\small{\rm{\frac{1}{x}}}$ has many applications in science and engineering.
In
the next
section, we will see a few more common functions.
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