[MUSIC] For a given value of x, it is not always possible to determine a unique value of y. The set x and y given y less than or equal to x, consists of the ordered pairs as (1,0) (1,1) and (1,-4). In this set, belong all points which satisfy the inequality y less or equal to x. From the graphical point of view, two or more points of a relation may belong to a single vertical line in the xy plane. If instead, for each value of x, there exists only one corresponding value of y, this relation is said to be a case in point. In this case, y is said to be a function of x. And this is denoted by y = f(x). Where f(x) does not mean f times x. These makes us to understand that a function is a set of ordered pairs. The main property is that, for any value of x, we have a unique value of y. It is true that a function must be a relation. However, a relation may not be a function. Moreover, a function requires a unique y for each x. The converse is not required. This implies that more than one x value, may be related to the same y value. In the function y = f(x), x is referred to as the argument of the function, y is called the value of the function. Or we can define x as the independent variable and y as the dependent variable. The set of all admittable values that x can take in a given range is called the domain of the function. This might be a subset of the set of all the real numbers. The value of y, into which an x value is traced, is called the image of x. The set of all images constitute the range of the function. This is the set of all values that the y variable can take. Therefore, we can say that while the domain refers to the independent variable x, the range instead refers to the dependent variable y. For example, total cost C per day is a function of the daily output Q in a firm such that C = 150 + 7Q. The capacity limit is of 100 units of output per day. Find the domain and the range of the cost function. If Q can vary only between 0 and 100, the domain is the set of values for Q higher or equal than 0, and Q lower or equal to 100. Let us now see what is the range. If we plot the function, we have a straight line with the minimum C value at 150 when Q = 0. In the maximum, C value at 850 when Q = 100. Then we have that the range is equal to the set of all values for C higher or equal to 150 and C lower or equal to 850. [MUSIC]