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ISLAMIYA ENGLISH SCHOOL, ABU DHABI. GRADE: 10 FUNCTONS CHAPTER: 3 RELATIONS: Sets are often related to each other. The relationship can be shown in many ways: 1. One to one. 2. Many to one. 3. Many to many. 4. One to many. Sometimes the relationship can be more complex. Complex relationships are usually shown more clearly on mapping diagrams. Example: A headed table. A mapping diagram using sets. A table. Ordered pairs. A mapping diagram using number lines. A graph. A box diagram. An algebraic relationship. In the above example the relationship is one-to-one because there is only one connection between the members of each set. FUNCTONS: ‘One to one’ and ‘many to one’ relations are called functions. A mapping diagram makes it easy to decide if a relation is a function or not. Whether a relation is a function can depend on the members of the sets involved. When deciding if a relation is a function you need to know what sets are involved. A mapping diagram can be used if the sets involved are small. If the sets involved are infinite then the vertical line test on a graph is used. A function is a set of rules for turning one number into another. Functions are very useful, for example, they are much used in computer spreadsheets. In effect a function is a computer, an imaginary box that turns an input number into an output number. Domain and range: One way to picture a function is as a mapping from one set to another. In this example the only numbers the function can use are from the set {1, 2, 4,7}. This set is called domain of the function. The set {3, 4, 6, 9} produced by the function 𝑓(𝑥) = 𝑥 + 2 is the range of the function. If the set {3, 4, 6, 9, 11, 13} on the right contains numbers with no arrows going to them, then this set is called the co-domain. The range is a subset of the co-domain. SAF MATHS DEPARTMENT 1 ISLAMIYA ENGLISH SCHOOL, ABU DHABI. The co-domain is {3, 4, 6, 9, 11, 13}. The range is still {3, 4, 6, 9}. Composite function: When one function is followed by another, the result is a composite function. KEY POINTS: 1. If only one arrow leaves the members of the set on the left then the relation is a function. 2. If a vertical line placed anywhere on a graph of a relationship intersects the graph at only one point then the relationship is a function. 3. A function can be written in three forms :(i) 𝑦 = 5𝑥 − 2 [𝑥, 𝑦 form]. (ii) 𝑓(𝑥) = 5𝑥 − 2 [bracket form]. (iii) 𝑓: 𝑥 ↦ 5𝑥 − 2 [arrow form]. All the three forms have the same meaning. 4. When a function is written in arrow form, the first letter is the name of the function, the letter between colon and arrow sign is domain ( x ), the part after the arrow is range ( y ). 5. To find 𝑓(2), replace x of f function by 2. 6. Composite function :(i) To find 𝑓𝑔 𝑜𝑟 𝑓𝑔(𝑥), replace x of f function by g or g(x) function. Except x, everything of f function will remain the same. (ii) To find 𝑓𝑔(2), first find 𝑔(2), then find 𝑓[𝑔(2)]. 7. If original function is f, inverse function is 𝑓 −1 . 8. To find inverse of f, follow the following steps :(i) Write f function in x, y form. (ii) Replace x by y and y by x. (iii) Make y the subject of the formula. (iv) Write 𝑓 −1 in the required form. 9. If in f function 𝑥 > 𝑎 and 𝑦 > 𝑏, in 𝑓 −1 function 𝑥 > 𝑏 and 𝑦 > 𝑎. 10. In a function involving a fraction , the value of x for which the denominator becomes zero, must be excluded. 1 11. For example, if 𝑓(𝑥) = 𝑥−2 , 𝑥 = 2 must be excluded, i.e. 𝑥 ≠ 2. 12. X values of a function are called domain and y values are called range. 13. To find the range of a quadratic function, if domain is not given :(i) Find the maximum or minimum value of y. (ii) If y is maximum, range is 𝑦 ≤ maximum value. (iii) If y is minimum, range is 𝑦 ≥ minimum value. 14. To find the range of any function, if domain is given :(i) Draw a sketch of the function for the given domain. (ii) Find the range from the sketch. x-----------------x---------------x SAF MATHS DEPARTMENT 2