A rational number is a number which can be expressed as a ratio of whole numbers. Technically every integer is a rational number because they can be expressed in the quotient form. Any rational number is a trivial example of an algebraic number.
The general form of rational numbers: A rational number is that which can be expressed as a/b where ‘a’ and ‘b’ are two different integers and where b is not equal to zero. Hence a number that is the quotient or ratio of two integers, and the denominator is not equal to zero, is called a rational number.
A rational function is a function which can be expressed as the quotient or ratio of two polynomial functions, where the polynomial in the denominator must have a degree of at least 1.
Rational functions provide notable examples and occur in many contexts naturally and are useful for practical approaches like the visual representation of a problem.
The domain R(f) = A(f)/B(f) is the set of all points ‘f’ for which the denominator is B(f) is not zero.
[Domain is the set of all values (f) over which the function is defined].
An example of a rational function is: R (f) =a +12a^(2−a) −1.
Note: One can determine the domain restrictions of a rational function by setting the denominator equal to zero and then solving.
Singularities are the f-values at which the B(f) or the denominator equals zero, and it is not a part of the domain of the function.
A rational function R (f) is the function in form A(f)/B (f) where A (f) and B (f) are polynomial functions and B (f) is a non-zero polynomial (A zero polynomial is a polynomial with zero value like 0a +0b +0).
The rational function formula is, R(f) =A(f)B(f)R(f) = A(f)B(f), where B(f)≠0.
Note: Every polynomial function is a rational function, but a rational function will only become a polynomial function when B (f) = 1, i.e., a constant polynomial function.
Also, according to the rational function definition, a function that cannot be expressed in the form of a polynomial, such as R(f) = sin(f), is not a rational function. However, a constant function such as R(f) = pi is a rational value function since constants are polynomials. The function itself is rational, even though the value of R(f) is irrational for all f-values.
Before representing a given rational fraction through the graph, one must take care of the values that potentially can make the denominator value zero and eventually can make the y-axis value undefined. These values are called excluded values for they must be excluded from the domain of the rational function.
A line which forever approaches the curve but never touches it is called an asymptote of a curve. Three kinds of Asymptotes are responsible for forming the types of rational functions, namely Horizontal Asymptote, Vertical Asymptote, and Oblique Asymptote.
A rational function can have many vertical asymptotes but will have at most one horizontal or oblique asymptote. Vertical asymptotes occur at singularities of a rational function, or on undefined points of the function. Still, the vertical asymptote only occurs at singularities where the associated linear factor in the denominator remains after cancellation. And the existence of an oblique or horizontal asymptote depends on the degrees of polynomials in the numerator and denominator of the rational function.
Types of rational functions depend on the kinds of Asymptote present in the graphing of rational functions.
Horizontal Asymptote is parallel to the axis of the independent variable (parallel to the x-axis). And horizontal asymptotes are horizontal lines that the graph of the rational function approaches as ‘f’ tends to +infinity or –infinity.
Vertical lines near which the function can grow boundlessly are called Vertical Asymptotes, and these lines are parallel to the y-axis.
Oblique Asymptote is one that is neither vertical nor horizontal asymptotes. The oblique or slant asymptotes are actually the diagonal lines in the curve that approaches zero as ‘f’ tends to +infinity or –infinity.
A rational function can have an oblique asymptote if the degree of the polynomial in the numerator (p) and the degree polynomial in the denominator (q). Then there can exist three possible cases,
If p>q, then there is no horizontal asymptote, but if p = q +1 then an oblique asymptote will exist.
If p<q, then the x-axis will be a horizontal asymptote.
If p=q, then there will exist a horizontal asymptote, and the equation will be:
y = (Coefficient of highest power term in the numerator) /(Coefficient of highest power term in the denominator).
On graphing a rational function, we get a horizontal asymptote (y=0), vertical asymptote (x=0), and the oblique or slant asymptote (blue line).
An equation containing at least one fraction with a polynomial numerator and denominator is called a rational equation. In the rational equations, these fractions might be situated on one or both sides. A common way to solve these equations is by reducing the fractions to a common denominator and then addressing the numerators’ equality.
A simple example problem can explain the process of solving the rational equations by multiplying with the least common denominator,
5/a −1/3 =1/a or, 3a(5/a –1/3) = 3a(1/a) or, 15 –a = 3 or, a = 12
Hence, we can see that the process is not very difficult.
An easy way to simplify a rational expression is by determining the domain, and to find the excluded values (those which result to form a zero denominator). Rational fractions can be simplified by removing terms that can be factored out of the numerator and denominator, and these can be either numbers or functions. The rational functions can also be multiplied together as the numerators of each polynomial can be multiplied together, as well as their denominators, and sometimes it will also be possible to simplify the resulting fraction. A rational fraction can be divided by one another, and this follows the standard rules for dividing fractions, where the dividend is multiplied by the reciprocal of the divisor.