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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:
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‘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:
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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
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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.
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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.
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SAF
MATHS DEPARTMENT
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