 Hello and welcome to the session. In this session we will discuss function notation and how to write domain and range of functions in intervals or using inequalities. Consider the statement the cost of gasoline is $3.50 per gallon. We can find the equation representing the situation. Let y be the cost and x represents number of gallons. Then according to the statement the equation will be y is equal to $3.50 into x. If we put values of x we will obtain the corresponding values of y. So x is input and y is output. Now if we draw the graph of this equation we will get a straight line. Here is the required graph of the equation that is y is equal to $3.50 into x and this line passes through the points that is $00, $1, $3.5, $2, $7, $3, $10.5 and so on. By vertical line test we see that the vertical line intersects this line exactly at one point so the given graph is a function. Hence this equation represents a function showing relation between x and y. Thus we can represent a function with the help of an equation and its notation is given by y is equal to f of x. For f of x we say y is equal to f of x that is y is a function of x also. Value of y depends on x so x is independent variable and y is dependent variable. Suppose we have a function f of x is equal to 3x minus 1 or we can say y is equal to 3x minus 1. This function serves as a function machine that is f converts x into 3x minus 1 that is if we put the value of x as 4 in f of x it will be converted into 3 into 4 minus 1 which is equal to 12 minus 1 that is 11. So we say that x is the input and y is the output. If we put any value of x we will obtain the corresponding value of y. For example let f of x be equal to 4x square minus 3x plus 1 and if we have to find the value of f of minus 2 to find the value of f of minus 2 we put x is equal to minus 2 in this function and we get f of minus 2 is equal to 4 into minus 2 whole square minus of 3 into minus 2 plus 1 which is equal to 4 into minus 2 whole square that is 4 minus of 3 into minus of 2 is plus 6 plus 1 which is further equal to 4 into 4 that is 16 plus 6 plus 1 that is 16 plus 6 plus 1 that is 23. So we say that f of minus 2 is equal to 23 it indicates the point with the coordinates minus 2 23 on the graph of f of x let us take another example. Let f be a function from set x to set y defined as f of x is equal to x square plus 1 then we have to write in simplest form the function f of x plus 1 and we also need to find its domain and range. Now here we are given a mapping f from set x to set y defined by the function f of x is equal to x square plus 1 and we have to find f of x plus 1. So here we replace x by x plus 1 in the given function and therefore we get f of x plus 1 is equal to x plus 1 whole square plus 1. Now using the identity a plus b whole square is equal to a square plus 2ab plus b square we write x plus 1 whole square as x square plus 2 into x into 1 that is 2f plus 1 square that is 1 plus 1. Which is equal to x square plus 2x plus 1 plus 1 that is plus 2 plus f of x plus 1 is equal to x square plus 2x plus 2. Now we have f of x is equal to x square plus 1. Here x can't take any real number value. So domain will be equal to set of all x such that x belongs to the set of real numbers are for range we have to see what values y can take or f of x can take. When we put any value of x we will get value of y as output since x is any real number it can be negative 0 or a pivotal real number. So when we put x as negative number say x is equal to minus of 1.2 and y is given by x square plus 1 and if we put the value of x as minus of 1.2 we will get the value of y as minus of 1.2 whole square plus 1 which is equal to minus of 1.2 whole square is 1.44 plus 1 which is equal to 2.44 that is for x is equal to minus of 1.2 y is equal to 2.44. Here we see when we square x it becomes positive and then 1 is added that we say that when x is negative then also y is positive. Now when x is equal to 0 we see that y will be equal to 0 square plus 1 which is equal to 0 plus 1 that is 1. It means the minimum value that y can take is 1 and when x is positive then y will also be a positive real number greater than 1 since x is positive therefore x square will also be positive and if we are adding 1 to it then we will get a positive real number greater than 1 thus we see that value of y that is f of x will never be negative and will be a real number equal to 1 or greater than 1 thus we say that y is greater than or equal to 1 so range that is r f will be equal to set of all y such that y is greater than or equal to 1. Now to conclude we can say that to find the domain of a function we need to consider what values of variable x make the function undefined for example consider the function y is equal to square root of x. Here square root of x has a meaning only when x is greater than or equal to 0 otherwise we cannot have a real value of square root of x when x is negative so we can take only positive values of x so domain of function f of x will be given by df is equal to set of all x such that x is greater than or equal to 0. Now let us consider another function f of x is equal to 1 upon x minus 2 we know that in rational expressions the function becomes undefined when denominator becomes 0. Here function becomes undefined when the denominator that is x minus 2 is equal to 0 that is when x is equal to 2 it means at x is equal to 2 f of x becomes undefined so x can take any real value except 2 so here domain of f of x will be equal to set of all x such that x belongs to the set of real numbers r and x is not equal to 2. So in this session we have discussed the function notation and how to write domain and range of a function in intervals or using inequalities. This completes our session hope you enjoyed this session.