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The A to Z of mathematics a basic guide - Sidebotham T.H.

Sidebotham T.H. The A to Z of mathematics a basic guide - Wiley publishing , 2002. - 489 p.
ISBN 0-471-15045-2
Download (direct link): theatozofmath2002.pdf
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Example 1. Verify that fix) = x2 + 2 is an even function.
Solution. Write
fix) = x2 + 2 f(-x) = (-x)2 + 2 = x2 + 2
fix) = f(~x)
Replacing a: by —a:
(-x)2 = x2
They are both equal to x2 + 2
Therefore f(x) = x2 + 2 is an even function. Example 2. Verify that f(x) = x3 is an odd function. Solution. Write
fix) = X3 f(-x) = (-xf
= —X3
fi-x) = - fix)
Replacing a: by —x
(—x)3 = —X3
They are both equal to —x~
Therefore fix) = x3 is an odd function.
References: Cosine, Cubic Equations, Function, Graphs, Parabola, Sine, Symmetry.
EVENT 183
EVENT
There are a number of terms in probability that can be explained together. In this entry the following terms are explained using examples: experiment, outcome, event, sample space.
William tosses two coins together a number of times, and records the results: This process is called an experiment. Another example of an experiment might be rolling two dice together, or drawing a card from a pack of 52 playing cards. In an experiment there are a number of possible outcomes and we have no way of predicting which outcome is next. The purpose of an experiment is to investigate the truth, or otherwise, of a statement. William is doing his experiment to investigate the truth, or not, that when two coins are tossed together a head (H) and a tail (T) are more likely to turn up than two heads. The sample space is a list of all possible outcomes, and for this particular experiment of tossing two coins together the sample space is given in the following table.
First coin H H T T
Second coin H T H T
Alternatively, the sample space can be expressed as HH, HT, TH, TT. We say there are four possible outcomes of this experiment.
An event is any subset of the sample space. When William tosses two coins together an event may be the coins landing the same side up, which is the subset HH, TT. Another event may be the coins landing differently, which is the subset HT, TH. Both these events are a subset, or part of, the sample space. An event may have only one outcome and then is known as a simple event. In our example, a simple event is tossing two heads, because there is only one possible outcome.
Example. An experiment is tossing three coins together. List the sample space, and list the event of obtaining two heads and one tail.
184 EXPANDING BRACKETS
Solution. A convenient way of obtaining a list of all the possible outcomes of this experiment, which is the sample space, is to use a tree diagram (see figure). The sample space is listed at the ends of the branches, and is stated here:
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
The event of tossing two heads and one tail is part of the sample space and is listed here:
HHT
HTH
THH
References: Complementary Events, Dice, Probability of an Event, Tree Diagram.
EXPANDED FORM OF DECIMALS
Reference: Compact form of Decimals.
EXPANDED FORM OF NUMBERS
Reference: Compact Form of Decimals.
EXPANDING BRACKETS
The process of expanding brackets results in an equivalent expression which contains no brackets. More simply, it is to rewrite an expression in order to remove the brackets. The resulting expression is called the expansion of the brackets, and is usually longer than the original expression. The explanation of expanding brackets will be a practical approach, but when the process is understood a simple algorithm will be
EXPANDING BRACKETS 185
used. Before proceeding further the reader should know how to multiply algebraic terms, as explained under the entry Algebra.
Example 1. Andrew has a square sandpit of side 3 m and wishes to extend it in one direction by 2 m (see figure a). Write down an expression for the total area of the new sandpit.
Solution. Write
Area of new sandpit = width x length
Area = 3 x (3 + 2) The length is enclosed in brackets
Area = 3x5 Area = 15 m2
Alternatively, we can consider the new sandpit as being made up of two smaller rectangles:
Area of new sandpit =3x3+3x2 Area = 9 + 6 Area = 15 m2
This example demonstrates that
3x(3 + 2) = 3x3 + 3x2
The expression on the right of the equals sign is written without brackets and is equivalent to the expression on the left of the equals sign. The expression on the right is the expansion of the bracket on the left. This process of expanding a bracket is known as the distributive law.
Brackets can be expanded using algebraic terms, as demonstrated in the next example.
Example 2. Suppose that Andrew did not know that the measurement of the original square sandpit was 3 by 3 and instead used x for its length and width (see figure b). Find an expression for the area of the new sandpit if it is extended by 2 m in one direction.
186 EXPANDING BRACKETS
X X2 2x
X 2
(b)
Solution. Write
Area of new sandpit = width x length
Area = x x (x + 2) Area = x(x + 2)
Replacing the original length of 3 by x In algebra the x sign is usually omitted
Alternatively, we can consider the new sandpit as being made up of two smaller rectangles:
Area of new sandpit = xxx-\-2xx
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