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Contents
Functions
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Introduction to Functions A very important type of relation between two sets A and B arises when each member of A is related to only one member of B. This relation is known as a mapping or a function. For the example below, set A = {-2, -2, 1, 2, 3} and set B is the set of all positive integers {1, 2, 3...}. The relation is "when squared equals". As each member of A has only one square, only one arrow leaves each number in A. Set A is called the domain and the part of set B is called the range. We say set A is mapped into set B. If we symbolise the the relation "when squared equals" by f, we can write f(A) = B. So f(-2) = 4 and 4 is a image of -2. Note: A function is always a relation but a relation is not necessarily a function. Summary: A relation f is a function from set A to set B when: 1) The entire set A if the domain of f. 2)f produced only one image in
B for each member of the domain.
Example 1A function f is defined
by f:x
2x2 + 1 with the domain X = {-1, 2, 3}. Find the image set of
f.
f(x) = 2x2 + 1 f(-1) = 2(-1)2 + 1 = 3
f(2) = 2(2)2 + 1 = 9
f(3) = 2(3)2 + 1 = 19
The image set is R = {3, 9, 19}
A
function f is defined by f:x
f(x) = x
i.e.
x2 + 5x - 5 = x
x2 + 4x - 5 = 0
(x + 5)(x - 1) = 0
x = -1 or 5
Since -5 is not in the domain, the value is 1.
A function f is
defined by f:x (a) the image of 5 under
f.
(b) the
possible values of x when the image is 8.
f(x) = 3x + 5/xf(5) = 3
x 5 + 5/5
= 16.
(a) The
image is 16.
 (b) The possible values of
x are 5/3 and 1.
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x2 + 5x - 5 for x>0. Find the value
of x which is unchanged by the mapping.
3x
+ 5/x, x 
0.
