Rational functions – Definition, Graph

Any function that can be defined by is called a rational function in mathematics. The numerator and denominator of a rational fraction are both polynomials. Polynomial coefficients do not have to be rational numbers; they can be in any field K. A rational function and a rational fraction over K are used in this case. The values of the variables can be found in any field L that contains K. The function’s domain is the set of variables’ values for which the denominator is not zero, and the codomain is L.

Properties of rational numbers

When a number can be expressed in the form p/q, it is considered a rational number, where both p and q are integers d q ≠ 0. . Rational numbers have 5 properties, which are listed below.

• Closure Property: The closure property of rational numbers states that when two rational numbers are added, subtracted, or multiplied, the result will be a rational number in all three cases.

• Commutative Property: The commutative property of rational numbers states that adding or multiplying two rational numbers in any order produces the same result.

• Associative Property: The associative property of rational numbers states that no matter how three rational numbers are grouped, the result remains the same when they are added or multiplied.

• Distributive Property: The distributive property of rational numbers states that any expression containing three rational numbers A, B, and C can be solved using the formula  A × (B + C) = AB + AC.

• Additive Property: If a/b is a rational number, the additive inverse property states that there exists a rational number (-a/b) such that a/b + (-a/b) = (-a/b) + a/b = 0.

Rational functions

“A rational function is a polynomial ratio in which the denominator polynomial is not equal to zero”.

A rational function is one in which the ratio of polynomials is the same. A rational function is one that has only one variable, x, and can be written as 

fx=pxq(x)

p(x) and q(x) are polynomials with q(x) ≠ 0.

Types of rational function

There are three types of asymptotes for a rational function: 

1. Horizontal Asymptote of a Rational Function

2. Vertical Asymptote of a Rational Function

3. Oblique Asymptotes of a Rational Function

 Horizontal Asymptote of a Rational Function

An imaginary horizontal line to which a function’s graph appears to be very close but never touches is called a horizontal asymptote (HA). It has the formula y = a number. “Some number” is closely associated with the range’s excluded values. A rational function can only have one horizontal asymptote. Using the degrees of the numerator (N) and denominators (D) is a simple way to find the horizontal asymptote of a rational function (D).

Vertical Asymptote of a Rational Function

A function’s vertical asymptote (VA) is an imaginary vertical line that its graph approaches but never touches. It has the formula x = a number. The domain’s excluded values are closely linked to “some number. “However, if there is a hole at x = some number, there cannot be a vertical asymptote at that number. One or more vertical asymptotes may exist for a rational function. To find the vertical asymptotes of a rational function, follow these steps:

Oblique Asymptotes of a Rational Function

A slant asymptote is an imaginary oblique line that appears to touch a portion of the graph. Only when the degree of the numerator (N) is exactly one greater than the degree of the denominator does a rational function have a slant asymptote (D). Its equation is y = quotient, which is obtained by using long division to divide the numerator by the denominator.

Example of a rational function

Find the x-intercepts of the rational function f(x) = (x2+ x- 2) / (x2– 2x- 3). 

The given function can be representation as:

fx=x+2(x-1)[(x-3)(x+1) 

It is not possible to cancel.

Substitute to find the x-intercepts.f(x)=0 

x+2x-1[x-3x+1]=0 

(x+2)(x-1)=0 

X=-2,x=1 

(-2, 0) and (1, 0) are the x-intercepts, 

Rational functions – Graph

Graphing a Rational Function Process If there are any intercepts, locate them. Remember that the y-intercept is (0,f(0)) ( 0 , f ( 0 ) )  and the x-intercepts are found by setting the numerator to zero and solving. Setting the denominator to zero and solving for the vertical asymptotes

f(x) = 1/x is the parent function of all rational functions and the simplest that can be written. Its graph is in fact a hyperbola curve, but not all rational function graphs are hyperbolas.

Conclusion

We conclude in this article that a “rational function” is a function that can be expressed as a quotient of polynomials, just as a “rational number” is a sum of whole numbers that can be expressed as a quotient. Rational functions are common in many situations and provide useful examples. Rational functions are used in a variety of ways in everyday life.