Introduction to functions

1. Introductionto functions For students of a third class C. Paganin I.T.I. Malignani - Udine

2. Aims What to Know • The definition of a function • The definition of a injective function • The definition of a surijective function • The definition of a bijective function • The properties of functions • The definition of an inverse function C. Paganin I.T.I. Malignani - Udine

3. What to do • Identify the domain and range of a function • Recognize the different forms of a function • Recognize graphically an injective function, a surjective function, a bijective function • Be able to compute the composite of two functions and identify its domain and range • Find the inverse function C. Paganin I.T.I. Malignani - Udine

4. Key Words for aims To recogize To use To calculate To find Be able to use rules for… to interpret the meaning… to describe growth… To recall To determinate To obtain To represent Be familiar with.. To apply the rules for… To carry out C. Paganin I.T.I. Malignani - Udine

5. Contents Students are asked to describe a few relations in some non-mathematical sets in order to appreciate that in some of them each element has one and only one corresponding element ( map ) E.G. In a first set , called A, they put all the students of the class and in a second set, called B, they write their shoe-sizes. Then they build a map between the two sets linking each student to his own shoe-size with an arrow. After considering some opportune examples, students can recognize three particular situations: • Any different elements of A have different images: the map is called injective • All the elements of B are linked to at least one element of A: the map is called surjective • The map is both injective and surjective, in which case it’s called bijective or ono-to-one. Then we pass to some mathematical examples: we consider a map between R and R (the set of real numbers) . We link an element x of the first set to an element y of the second set so defined ,e.g., y=3x+1. In this case the word “function” is more used than “map”, even if their properties are the same , and we usually write y=f(x). C. Paganin I.T.I. Malignani - Udine

6. The fist problem is to define the domain A of the function f. It is a subset of R and consists of the elements x involved in the function. Then we define the range B which consists of the elements y of the second set for which there is at least one x with f(x)=y. After many examples of different kinds of function and always involving students at the blackboard or working in pairs or in the whole class, they become familiar with injectivity and surjectivity and are ready to pass to bijectivity, that is to the inverse function of a given function. At the beginning students are asked to draw the function f(x)=y before considered y= 3x+1 in a sheet, then they write the inverse function of it y=(x-1):3, obtained interchanging x and y, and draw also the inverse function in the same sheet;after that they fold the sheet along the bisector line of the first and third quadrants and be aware that the two graphs overlap! C. Paganin I.T.I. Malignani - Udine

7. Conclusion Students are asked to be protagonists and they enjoy a lot … even if it’s more difficult to sleep during Maths lessons! C. Paganin I.T.I. Malignani - Udine