[MUSIC] Functions are important in several areas of studies. In particular, they are very useful tools in mathematics applied to economics, finance, and business. The economic analysis is plenty of terms like demand and supply functions. Also we have the cost function, and again the production function, and finally the consumption function. One variable is said to be a function of another if the first variable depends upon the second. As we have seen, a function is a correspondence between two sets of elements. It is important that to each element in the first set, their corresponds one and only one element in the other set. The first set is called the domain, and the second set is called the range. We define the variable which represents the elements in the domain of a function, for instance, x, the independent variable. It is also referred to as input. The variable which represents the elements in the range, for instance, y, is called dependent variable. It is also referred to as output. The graphical representation is very important because helps as to visualize the function. This is because the shape of the graph reflects the properties of the function. In order to sketch the graph of an equation in x and y, we need to find the ordered pairs that solve the equation. After, we will plot these ordered pairs. The above process is called point-by-point plotting. We sketch the graph on the Cartesian plane, which is a coordinate system obtained by drawing two perpendicular lines. These in turn are called coordinate axis. We have the horizontal axis which is called the x-axis and the vertical axis which is called the y-axis, The intersection point between the two is called the origin, that is defined with O. The Cartesian plane is also called the xy-plane. The four quadrants normally are numbered. For example, let's the plot the graph of the equation. y = x to the power of 2 + 2. Given an equation's variables, if we get exactly one output that is the value for the dependent variable for each input. Which is the value of the independent variable. Then the equation specifies the function. This is why, in that case, the graph specifies the equation. However, in the case that we get more than one output for a given input, dismiss that the equation does not specify a function. Consider our equation, y = x to the power of 2 + 2. If (-2,4) is another pair of the function, our input is x = -2. We need to square -2, which is equal to 4, and then we add 2. Then our output is equal to 6. Consider again our function, f(x) = x to the power of 2 + 2. If x is equal to -3, what does it mean? We substitute the value -3 in x, and then we have f(-3) = -3 times 2 + 2. We square -3 which is equal to 9. Then we add 2 and the output is 11. Therefore the point (-3, 11) is on the graph of the function. Let's make a table of the ordered pairs that satisfy the equation. If x is equal to -3, then we have (-3)2 + 2 = 11. If x is equal to -2, then we have (-2)2 + 2 = 6. If x is equal to -1, then we have (-1)2 + 2 = 3. If we have that x is equal to 0, then we have (0)2 + 2 = 2. If x instead is equal to 1, then we have (1)2 + 2 = 3. If x is equal to 2, then we have (2)2 + 2 = 6. In our example, we have a polynomial function. In particular, it is a quadratic function. Therefore, if we plot the points and connect them, we will end up with a parabola. Since in our case, a is greater than 0, therefore, we have an upwards move curve. [MUSIC]