What are Rational Functions?

A rational function is a type of function which is in form of f(x) = \frac { P(x) }{ Q(x) }, where Q(x) \neq 0. In simple words, a rational function is a function that is written in a fraction. The Q(x) can be a single number or a polynomial. Hence, the criterion is given by a quotient between polynomials:

f(x) = \frac { { a }_{ 0 } + { a }_{ 1 }x + { a }_{ 2 } { x }^{ 2 } + ... { a }_{ n }{ x }^{ n } }{ { b }_{ 0 } + { b }_{ 1 }x + { b }_{ 2 } { x }^{ 2 } + ... { b }_{ m }{ x }^{ m } }

The domain is equal to \mathbb{R}, minus the values of x that would annul the denominator. What if the Q(x) = 1? Do we still call it rational function? The answer is yes! It also means that every polynomial function is a rational function, how? Make Q(x) = 1 and hence it will become a rational function.

For example, f(x) = \frac { { x }^{ 2 } + 5x + 4 }{ x - 1 }.

We are interested in the denominator because the problem with rational function is that they tend to go to infinity at a point. That is why we make a range for that specific function. The denominator of the above function is x - 1. Make the denominator equal to zero.

x - 1 = 0 \qquad x = 1

It means that when x will be equal to 1, the function will reach infinity. Hence we can say that the domain of this function is all the values of x except 1.

The best Maths tutors available
Intasar
4.9
4.9 (36 reviews)
Intasar
£48
/h
Gift icon
1st lesson free!
Paolo
4.9
4.9 (28 reviews)
Paolo
£30
/h
Gift icon
1st lesson free!
Shane
4.9
4.9 (23 reviews)
Shane
£25
/h
Gift icon
1st lesson free!
Jamie
5
5 (16 reviews)
Jamie
£25
/h
Gift icon
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£30
/h
Gift icon
1st lesson free!
Petar
4.9
4.9 (12 reviews)
Petar
£40
/h
Gift icon
1st lesson free!
Farooq
4.9
4.9 (17 reviews)
Farooq
£40
/h
Gift icon
1st lesson free!
Dr. Kritaphat
4.9
4.9 (7 reviews)
Dr. Kritaphat
£49
/h
Gift icon
1st lesson free!
Intasar
4.9
4.9 (36 reviews)
Intasar
£48
/h
Gift icon
1st lesson free!
Paolo
4.9
4.9 (28 reviews)
Paolo
£30
/h
Gift icon
1st lesson free!
Shane
4.9
4.9 (23 reviews)
Shane
£25
/h
Gift icon
1st lesson free!
Jamie
5
5 (16 reviews)
Jamie
£25
/h
Gift icon
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£30
/h
Gift icon
1st lesson free!
Petar
4.9
4.9 (12 reviews)
Petar
£40
/h
Gift icon
1st lesson free!
Farooq
4.9
4.9 (17 reviews)
Farooq
£40
/h
Gift icon
1st lesson free!
Dr. Kritaphat
4.9
4.9 (7 reviews)
Dr. Kritaphat
£49
/h
Gift icon
1st lesson free!
Let's go

Asymptotes

Taking about rational function, the biggest use is in finding asymptotes of the hyperbola. An asymptote is a line that is drawn beside the curve but they both don't touch each other unless they reach infinity. Asymptotes are categorized into three types, horizontal, vertical, and oblique.  The functions of the type y = \frac { k }{ x } has a hyperbola in its graph, however, they look something like this y = \frac { ax + b }{ cx + d }. The simplest hyperbola curve equation is represented like this: y = \frac { k }{ x } with its asymptotes as axes and the center of the hyperbola is the origin (it is also the point where both asymptotes intersect).

For example, f(x) = \frac { 2 }{ x }

Vertical Translation

In this translation, the vertical asymptote will be shifted according to the equation. In vertical translation, the abscissa will always be zero. Here is the syntax of the equation:

f(x) = \frac { k }{ x } + a

The center of the hyperbola is (0, a). If a > 0, y = \frac { k }{ x } moves upward "a" units. For example:

y = \frac { 2 }{ x } + 3

The center of the hyperbola is: (0, 3)

If a < 0, y = \frac { k }{ x } moves downwards "a" units. For example:

y = \frac { 2 }{ x } - 3

The center of the hyperbola is: (0, -3)

Horizontal Translation

For the case of horizontal translation, the asymptote shifts horizontally according to the equation. In horizontal translation, the ordinate will always be zero. Below is the syntax of the equation:

y = \frac { k }{ (x + b) }

The center of the hyperbola is: (-b, 0). Do note the minus sign in the center coordinates of the hyperbola, this is a very common mistake.

If b > 0, y = \frac { k }{ x } is shifted to the left "b" units. For example:

y = \frac { 2 }{ x + 3 }

The center of the hyperbola is: (-3, 0)

If b < 0, y = \frac { k }{ x } is shifted to the right "b" units. For example:

y = \frac { 2 }{ x - 3 }

The center of the hyperbola is: (3, 0)

Oblique Translation

The oblique translation consists of two translation at the same time. The horizontal and vertical translations. Here is the syntax of the equation:

y = \frac { k }{ (x + b) } + a

Neither of the coordinates can be zero under the oblique translation condition. The center of the hyperbola is (-b, a). For example:

y = \frac { 2 }{ x - 3 } + 4

The center of the hyperbola is: (3, 4).

Graphing Hyperbolas

Remember, we told you that the hyperbola equation looks something like this in reality:

y = \frac { ax + b }{ cx + d }

It is modified and then written as

y = \frac { k }{ (x + b) } + a

Where "a" and "b" determines the point where both asymptotes meet each other. Its graph is a hyperbola with a center (−b, a) and asymptotes parallel to the axes.

Example

y = \frac { 3x + 5 }{ x + 1 }

y = \frac { 3x + 3 + 2 }{ x + 1 }

y = \frac { 3x + 3 }{ x + 1 } + \frac { 2 }{ x + 1 }

y = 3 + \frac { 2 }{ x + 1 }

Compare it to y = \frac { k }{ (x + b) } + a

a = 3 \qquad b = 1

Remember, the center is written as: (-b, a)

The center of the hyperbola is: (-1, 3).

>

The platform that connects tutors and students

Did you like this article? Rate it!

1 Star2 Stars3 Stars4 Stars5 Stars 5.00 (2 rating(s))
Loading...

Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.