We know that all the mathematical operations can be performed on the Rational Numbers. We say that the Rational expression is the Mathematical Expression in which we can find the rational numbers combined together with the arithmetical operators. Lets learn about the rational expressions and the method used to simplify rational expressions.
Graphing Rational FunctionBack to Top
Suppose we have a rational function f (p) = p2 - 4 / p2 – 4p, then we can plot graph of rational function as shown below.
Solution: - Here we need to follow some steps to plot graph of rational function.
Step 1: First we check whether function is rational function or not. As we know that ratio of two polynomials is known as rational function. So rational function is:
=> f (p) = p2 - 4 / p2 – 4p,
Here in this function we cannot put value of denominator as zero because if we put value of denominator as zero then whole function changes to infinity. There is no y – intercept present for graph.
Step 2: Put numerator value equals to zero to get value of x – coordinate.
So we can write numerator value as:
=> p2 – 4 = 0, so here we get two values of p that is p = + 2.
Above function can also be written as:
= p2 – 4 = p (p – 4) = 0, so here we get value of 'p' as 0 and 4.
Step 3: Now put different values of 'p' to get other values.
If we put value of 'p' as 1, 3 and 5 then we get 1, -5/3 and 21/5 respectively. Now we can easily plot graph of rational function using these coordinates. Graph of rational function is shown below:
Add and Subtract Rational ExpressionBack to Top
Let’s consider a function f (y) which is represented as:
f (y) = P (x) / Q (x) where P (x) and Q (x) are two polynomials. Function f (x) is a Rational Expression. f (y) = sin y is not a Rational Function. Here polynomial in denominator cannot be zero that is B (y) ≠ 0.
Let’s see how to add subtract rational expressions. Following steps are used:
1) First of all, factor all the values and get the Least Common Denominator.
2) Then multiply the numerators and denominators with such a denominator that makes the denominator common.
3) Add the numerators together by taking L.H.S.
4) Factor the numerator.
5) Finally, cancel all common factors.
Let’s take an example to understand the concept of addition and subtraction of rational expressions.
3y + 1 / (y2 – 1) + y / (y + 1) = 3y + 1 / (y + 1) (y– 1) + y / (y + 1), least common denominator is
(y + 1) (y – 1).
Then [3y + 1 / (y + 1) (y– 1)] + [y / (y + 1) (y – 1) / (y – 1)],
= [(3y + 1) + y (y – 1)] / [(y + 1) (y– 1)],
= [(3y + 1 + y2– y)] / [(y + 1) (y– 1)],
= [(y2 + 2y +1)] / [(y + 1) (y– 1)],
Fraction of numerator part will be (y + 1)2,
Then final solution will be,
3y + 1 / (y2 – 1) + y / (y + 1) = (y + 1) / (y – 1).
Inverse VariationBack to Top
Suppose we have two variables: “a” and “b” such that,
1. Value of “b” differs inversely as that of “a”,
2. “b” is inversely proportional to “a”.
Actual relation can be written as:
b = f a, where 'f' is some fixed real number and not equals to zero. “f” is also known as variation constant or proportionality constant.
Let us understand the formula for Inverse Variation:
Connection that we just saw between two variables, will always give multiplication as a constant. There are certain steps which have to be followed in order to retrieve the inverse variation relation among the variables. We have to interpret the statement in form of an inverse variation relation.
Step 1. According to 2nd Point as shown above, “b” is inversely proportional to “a” values which means:
b = f x,
Step 2. Next put the known values i.e. “a” and “b” in the equation, we wrote in step 1 to get the value of non - zero real number 'f'.
Step 3. Lastly, put the value of 'k' in the original equation to get the desired equation.
Let us consider some examples of it.
Example: If b = 20 when a = 4, then write the formula for relation between 'a' and 'b' given 'b' is inversely proportional to 'a'?
Step 1. Convert the given statement to an inverse variation formula. In our example we will write it as follows: “b” is inversely proportional to “a” means
b = f * a,
Step 2: Now substitute the known values i.e. at a = 4 we have b = 20. To get the corresponding value of 'f' as follows:
20 = f * 4,
Or f = 5,
So, general relation that we can write using value of 'f' as:
b = 5a.
Indirect VariationsBack to Top
So, if we increase our speed by twice its value, then time decrease by two times. We can say that product of two variables in indirect variation remains constant as compared to direct variations, where ration of two quantities remains constant.
General expression for indirect variation or indirect proportionality can be given by considering two variables 'n' and 'm'. 'n' is inversely proportional to 'm' and which is given as follows:
N * m = k or n = k / m,
Let us consider an example of indirect variation to understand it better.
Example: If a car has an average speed of 80 miles per hour and it takes 7 hours to complete its journey, then how long it would take to complete the same journey with a speed of 100 miles per hour?
Solution: Let speed be denoted by variable 's' and time by 't'. Distance can be calculated as follows:
D = s * t = 80 * 7 = 560 miles,
To calculate the time taken to complete the same distance at a speed of 100 miles per hour
t = 560 / 100 = 5.6 hours.
This shows that speed and time are in indirect relation and thus are inversely proportional to each other. We saw that at speed 80 miles / hour, time taken was 7 hours and at speed 100 miles / hour, time taken is 5.6 hours.
Direct VariationsBack to Top
For instance, suppose we have two variables “a” and “b” such that they are in a relation a = 4b i.e. value of “a” will always be four times the corresponding value of “b” or we can say that the values of “a” and “b” are in a fixed Ratio 1 / 4. That is, if value of 'b' is 4 then a’s value will be 16 and similarly if a’s value is 20, b’s value will be 5.
To define the direct variations more specifically, let's say there are two variables A and B such that A varies directly as B. When we plot the value of A and B, then we get a straight line with all pairs of coordinates lying on this line. Also this line would pass through the origin i.e. (0, 0). We can summarize as: Each variable is a constant multiple of every other number.
Direct variations can be expressed in form of equations as follows:
Suppose we have an equation m / n = 10. It states a direct variation between two variables m and n. Constant ratio that is maintained among the random values of m and n is 10.
Direct variations can also be represented as algebraic expressions.
For example, we have an expression given as: m = x n. Where, 'x' is a constant non zero factor that can be termed as a constant ratio maintained between the two variables 'm' and 'n'. For any random value of 'm' and 'n' we can calculate various values of 'x'. This can lead to many direct variations among different Sets of values of 'm' and 'n'.
Solving Rational EquationsBack to Top
As we studied above that rational equations contain terms in Ratio form. For Example: 4 / x + 7 = 2 / 3x is a rational equation because it has x in denominator. Let us see steps for solving rational equations:
Step 1: Assume that we have a rational equation 4 / x + 7 = 2 / 3x.
Step 2: Now find the Least Common Denominator of all the Fractions present in the rational equation. In above rational equation least common denominator is 3x.
Step 3: Now we will multiply all the terms by least common denominator.
(3x) 4 / x + (3x) 7 = (3x) 2 / 3x.
Step 4: Now just simplify all the terms of rational equation.
Step 5: We get.
12 + 21 x = 2
Step 6: Now we have a simplified equation. From this equation we can easily find the value of variable 'x'.
Step 7: Separate the variables and constant terms by moving them on either sides of equal sign.
21x = - 12 + 2,
21 x = -10,
X = -10 / 21
Multiply and Divide Rational ExpressionBack to Top
For multiplication we need two rational expressions. Rational expression has numerator and denominator. 'p' is numerator and 'q' is denominator. Let us discuss steps for multiplication and division of rational expressions:
Step 1: Assume that we have two Rational Numbers 2x /3 and 4y/ 5.
Step 2: For multiplication we multiply the numerator with numerator and denominator with denominator.
Step 3: When we multiply these rational numbers 2x /3 and 4y / 5 we will get 2x / 3 * 4y / 5 = 8xy / 15.
Step 4: Simplify the rational number.
Steps to find the division of rational expressions are:
Step 1: Division of rational expression is very similar to multiplication of rational expressions. In division of rational expressions we just reverse the second rational expression. Assume that we have two rational expressions (2x + 2) /(x + 1) and (2x + 1) / (x + 1).
Step 2: Now reverse the second rational expression as (x + 1) / (2x + 1).
Step 3: Follow the same procedure of multiplication.
Step 4: Multiply (2x + 2) /(x + 1) and (x + 1) / (2x + 1).
Step 5: Division of [(2x + 2) /(x + 1)] * [(x + 1) / (2x + 1)].
Step 6: Result of division will be 4x2 + 6x + 2.
These are the steps to multiply and divide rational expressions.