Rational Function & Rational Number

Introduction

The canonical form of rational numbers can be defined, when there is no common element other than 1 between the dividend and the divisor, and therefore the divisor is positive.

There is only one factor in common between divisors and dividend. Therefore, it can be said that rational numbers are $\mathrm{\frac{1}{3}}$ in the canonical form.

Rational Numbers

To determine if a number is a rational number, check the following conditions: This is expressed in the form of $\mathrm{\frac{p}{q}}$, where q ≠ 0. The ratio $\mathrm{\frac{p}{q}}$ has been further simplified and can be expressed in decimal format.

Rational Functions

A rational function is the ratio of polynomials whose denominator polynomial must not be zero. Did you know that rational functions are used in various areas of our daily lives?

Example, $\mathrm{g (x) =\frac{(x^2 + x^{-2})}{(2x^2-2x^{-3})}}$ is a rational function, where 2x²-2x^-3)≠ 0.

Everyone knows that a constant is a constant because it is a polynomial.

Graphing of Rational Functions

To graph a rational function −

• Dotted lines identify and draw vertical asymptotes.

• Dotted lines identify and draw horizontal asymptotes.

• Draw holes (if present).

• Plot all the points and then join them.

Asymptotes

There are three types of asymptotes in rational functions: horizontal, vertical, and diagonal asymptotes. Apart from that, it can also have holes. Let's see how to find each one.

Rational Function Hole

The rational function hole appears to exist in the rational function graph, but it does not actually exist. These can be obtained by setting the linear factor, which is the common factor of the numerator and denominator of the function, equal to zero and solving x. You can find the corresponding y-coordinate of a point by assigning an x-value in a simplified function. Not all rational functions need to have holes. Holes are only present if the numerator and denominator have a linear common divisor.

Vertical asymptote of rational function

The vertical asymptote (VA) of the function is a fictitious vertical line that the graph appears very close to, but never touches. x = can be in any number format. Here, "some numbers" are closely related to the values excluded from the domain. However, keep in mind that if x = number, there can be no vertical asymptote if there is a hole in the same number. Rational functions can have one or more vertical asymptotes. To find the vertical asymptote of a rational function −

First, simplify the function to eliminate common factors (if any).

Set the denominator to 0 and solve (x) (or equivalently get the value excluded from the domain avoiding the hole)

Horizontal Asymptote of Rational Function

Horizontal asymptote (HA) is a fictitious horizontal line that the graph appears very close to, but never touches. y = any number format. Here, "arbitrary number" is closely related to the value excluded from the range. An easy way to find the horizontal asymptote of a rational function is to use the order of the numerator (N) and denominator (D).

• If N then y = 0 has HA.

• If N> D, there is no HA.

• If N = D, HA is equal to the ratio of y = leading coefficients.

Diagonal asymptote of rational function

Diagonal asymptote is also an imaginary diagonal line that appears to touch part of the graph. The equation is the quotient of the numerator divided by the denominator using y = long division.

Comparison between Rational Numbers and Rational Functions

The terms "fraction" and "rational number" are closely related, but they differ in several respects. Note that "fractions are always rational numbers, but rational numbers may or may not be fractions."

Definition of Fractions and Rational Number

Fractions are arbitrary numbers of the form $\mathrm{\frac{a}{b}}$, where "a" and "b" are both integers and b ≠ 0.

Rational numbers, on the other hand, are numbers of the form $\mathrm{\frac{p}{q}}$, where "p" and "q" are both integers, q ≠ 0.

Therefore, fractions are written in the form $\mathrm{\frac{m}{n}}$. Where n is not 0 and m & n is an integer (or an integer).

Example: 23.12, 32.10. , 10.12. , 21.04. Rational numbers can be represented in the form $\mathrm{\frac{a}{b}}$. Where b is not 0 and a & b is an integer. For example, $\mathrm{\frac{1}{4},\frac{-9}{2},\frac{-12}{8}}$.

Rational Fraction

• Described in the format of rational fractions $\mathrm{\frac{a}{b}}$. Where a and b are integers and b ≠ 0.

• All fractions are reasonable.

• Example: $\mathrm{\frac{1}{2},\frac{1}{8},\frac{6}{4}}$ etc.

Rational Numbers

• Described in the format of rational numbers $\mathrm{\frac{p}{q}}$. Where $\mathrm{p,q\in Z}$, q ≠ 0.

• All rational numbers are not fractions.

• Examples of Rational Numbers are $\mathrm{\frac{1}{4},\frac{-9}{2},\frac{-12}{8}}$ etc.

Solved Examples

1.Find the inverse: $\mathrm{\mathit{f}(x) =\frac{(2x - 1)}{(x + 3)}}$

Solution:

Given,

$$\mathrm{y=\frac{(2x - 1)}{(x + 3)}}4$$

Interchanging x and y −

$$\mathrm{x=\frac{(2y - 1)}{(y + 3)}}$$

Now,

$$\mathrm{x(y + 3) = 2y - 1}$$

$$\mathrm{xy + 3x = 2y - 1}$$

$$\mathrm{3x + 1 = 2y - xy}$$

$$\mathrm{3x + 1 = y (2 - x)}$$

$$\mathrm{y =\frac{(3x + 1)}{(2 - x)} = \mathit{f}^{-1} (x)}$$

$$$$

$$$$

2. A function $\mathrm{\mathit{f}(x) =\frac{2 (x + 3)}{(x + 3)}+[\frac{1}{(x +3)}]}$ is given. Tell with reason whether it is rational or not.

Solution:

Here, the function is,

$$\mathrm{\mathit{f}(x) =\frac{2 (x + 3)}{(x + 3)}+[\frac{1}{(x +3)}]}$$

$$\mathrm{=\frac{2x + 7}{(x + 3)}}$$

$\mathrm{=\frac{p(x)}{q(x)}}$, here, both p(x) ,q(x)are polynomials.

Conclusion

Following integers, rational numbers are one of the most common types of numbers we learn in mathematics. "Ratio" is exactly what it is called. As a result, rational numbers have a strong connection to the concept of ratios.

Rational numbers are of the form $\mathrm{\frac{p}{q}}$, where p and q are both integers and q is nonzero. Q is an abbreviation for rational number.

Numbers in the $\mathrm{\frac{p}{q}}$ format make it difficult to distinguish between fractions and rational numbers. Fractions are made up of integers, and the numerator and denominator of rational numbers are made up of integers.

FAQs

1. Is the fraction a rational number?

All fractions are rational numbers, but not all rational numbers need to be fractions.

2. Differentiate between a rational fraction and a rational number?

Fractions are represented by the integer ratios a/b and b ≠ 0.

Rational numbers are represented by the ratio p/q. Where the numerator and denominator are integers, q ≠ 0.

3. What are examples of rational numbers and fractions?

Examples of rational numbers are $\mathrm{\frac{1}{4},\frac{-9}{2},\frac{-12}{8}}$, and so on. Examples of fractions are $\mathrm{\frac{1}{2},\frac{1}{8},\frac{6}{4}}$, and so on.

4. Give examples of good and bad fractions.

A good fraction is one whose denominator is larger than the numerator. For example, $\mathrm{\frac{1}{2},\frac{1}{3},\frac{1}{4}}$ etc. Inappropriate fractions are those whose numerator is larger than the denominator, for example $\mathrm{\frac{3}{4},\frac{5}{4},\frac{7}{6}}$ etc.

5. Which of the two rational numbers $\mathrm{\frac{2}{5}}$ or $\mathrm{\frac{-3}{4}}$ is larger?

Given rational numbers are $\mathrm{\frac{2}{5}}$ and $\mathrm{\frac{-3}{4}}$. We clearly know that between positive and negative rational numbers, the positive rational number is always large. Therefore, $\mathrm{\frac{2}{5}}$ is greater than $\mathrm{\frac{-3}{4}}$.