 Hello and welcome to the session. In this session we will discuss the domain and range for rational functions. And we will also learn to simplify rational expressions. First of all we are going to discuss rational function. A rational function is a function of the form f of x upon g of x and g of x is not equal to 0 and f of x and g of x are polynomials in x. For example, x plus 1 whole upon 2x plus 3 is a rational function. We know that domain of a function is the set of possible values of the independent variable that is x values for which the given function is defined. Once we know the domain of the function we can determine the y values and the set of possible values of the dependent variable that is y values is called range of the function. For the rational function of the form f of x upon g of x where g of x is not equal to 0 we can find the domain by using the following steps. The first step is to set the denominator of rational function equal to 0 that is g of x is equal to 0. Then we solve this for x. Now we obtain values of x in the above steps make the function undefined so all the values other than these values are domain of the rational function. Graphically this type of exclusion from the domain is usually where there are vertical SM totes. Division by 0 results in no y value output. The graph just goes to infinity or negative infinity on either side of the SM totes. It's also possible to have a hole in the graph instead of an SM totes but the function is still undefined at the hole and the corresponding x value must still be excluded. So basically we have to check if there are any x values that results in a division by 0 and therefore makes the function undefined. To determine the range we look at the nature of the function to see if there are any restrictions on the y values restrictions on the domain may or may not restrict the range. The domain may be restricted but the range is not or the domain may be all real numbers but the range is restricted. And we should note that it is extremely helpful to have some idea about how the function looks when graphed. Let us consider an example. Find domain and range of the function f of x is equal to 5 upon 2x minus 3. Here we are given this rational function. So first of all we will find its domain. To find the domain we will set its denominator equal to 0 that is here denominator is 2x minus 3 so we set this equal to 0. From here we get the value of x as 3 by 2 that is equal to 1.5. So for x is equal to 1.5 the function becomes undefined hence this point should be excluded from the domain. Thus domain consists of all real numbers except 1.5. Hence domain is given by the set of all x where x belongs to the set of real numbers and x is not equal to 1.5. All we can also write it as domain is equal to open interval from minus infinity to 1.5 union open interval from 1.5 to infinity. Now we have got the domain. From this domain if we choose any value of x except 1.5 and put it in the given function then we see that as x becomes large f of x becomes very small but it never becomes 0. So range of the function will be the set of all real numbers except 0. So we can write range is equal to set of all f of x such that f of x belongs to the set of real numbers r and f of x is not equal to 0. Or we can also write it as the set of all y such that y belongs to the set of real numbers r and y is not equal to 0. In interval notation form we can say range is equal to open interval from minus infinity to 0 union open interval from 0 to infinity. Thus we have got the domain and range of the given function. Now let us see how to simplify rational expressions. A rational expression is said to be simplified if the numerator and denominator have no common factors other than one. Now we are going to discuss these steps to simplify a rational expression. The first step is we factorize the numerator and the denominator and then we cancel out the common factors to simplify the expression. Let us consider the following example. Simplify the rational expression 2x plus 4 whole upon x square minus 4. Let us start with its solution. So here first we factorize the numerator and denominator. In the numerator we take 2 common and we get 2 into x plus 2 the whole. And in the denominator we use the formula a square minus b square is equal to a plus b the whole into a minus b the whole. And here x square minus 4 can be written as x square minus 2 square and now applying this formula we get x plus 2 the whole into x minus 2 the whole. So here in the denominator we write x plus 2 the whole into x minus 2 the whole. Next we cancel out the common factors to simplify the expression. Here we see that the common factor is x plus 2 the whole. So here x plus 2 cancels with x plus 2 and we are left with 2 upon x minus 2. So we can write 2x plus 4 whole upon x square minus 4 is equal to 2 upon x minus 2 where x is not equal to 2. We should note that if we put the value of x as 2 in this rational expression then it will become undefined. Thus in this session we have learnt how to determine the domain and range for rational functions and we have also learnt to simplify rational expressions. This completes our session. Hope you enjoyed this session.