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

ISBN 0-471-15045-2

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We use inverse operations in solving equations by the “balancing method” and in finding angles in trigonometric problems.

References: Balancing an Equation, Inverse Trigonometric Ratios, Linear Equation, Trigonometry.

INVERSE PROPORTION

Reference: Proportion.

INVERSE RELATIONS

Suppose we have the relation “is older than” between three members of a family: Tom the grandfather, Joanne his daughter, and Luke his grandson. The arrow graph is shown in figure a, and the set of ordered pairs for this relation R is

If the arrow graph and the ordered pairs are reversed, we obtain the inverse relation of R, which has the symbol i?-1:

In words, the inverse relation is “is younger than.”

There is a rule connecting the graph of an algebraic function/with the graph of its inverse f~l, as illustrated in the following example.

Example 1. The graph of the function fix) = 2x — 1 can be drawn using the following ordered pairs (figure b):

R = {(Tom, Joanne), (Joanne, Luke), (Tom, Luke)}

R 1 = {(Joanne, Tom), (Luke, Joanne), (Luke, Tom)}

/ = {(-2, —5), (—1, —3), (0, -1),(1, 1), (2, 3)}

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265

The graph of the inverse function is drawn by reversing the ordered pairs off. This is achieved by exchanging x and y values:

/-*=(-5, —2), (—3, -1),(-1, 0),(1, 1),(3, 2))

In figure b the graphs of / and /-1 are drawn on the same axes. The rule connecting the graph of / and the graph of /_1 is that they are images of each other after reflection in the mirror line whose equation is y = x. The mirror line is drawn dashed in the figure. This rule is true for all graphs and their inverses.

The equation of the inverse function /-1 can usually be found by rearranging the equation of the function/, as demonstrated in the following example.

Example 2. If the equation of a function is fix) = 2x — 1, find the equation of the inverse function f~l.

Solution. Let the equation of the function be y = 2x — 1:

x = 2y — 1 The rule for finding inverses is to exchange x and y

The next step is to rewrite this equation as y =?, using a process known as changing the subject of a formula:

x + 1

x \

......

r1

2y

y

x + 1 ......

x + 1 ......

Adding 1 to both sides of the equation Dividing both sides of the equation by 2

Writing the equation with y on the left-hand side

The inverse of a function / is also a function if it has one-to-one correspondence.

References: Arrow Graph, Correspondence, Function, Mirror Line, Ordered Pairs, Relation.

266 ISOMETRIC

INVERSE TRIGONOMETRIC RATIOS

References: Inverse Operation, Trigonometry.

INVESTMENT

An investment is lending money to someone, or an institution like a bank, in order to make a profit. For example, Jo invests $1000 in a bank. After 1 year she makes a profit of 10% on her investment: 10% of 1000 = 100, so she has a profit of $100 on her investment.

References: Compound Interest, Interest, Simple Interest.

IRRATIONAL NUMBERS

The symbol for irrational numbers is Q;.

Reference: Integers.

ISOMETRIC

Sometimes we need to draw three-dimensional shapes on two-dimensional paper, but still make them appear as three-dimensional shapes. To do this, we can use isometric paper, which is made up of a regular pattern of dots, or spots. These dots are placed at the vertices of equilateral triangles, as shown in the figure. Isometric means equal measure. This paper is also known as isometric graph paper.

The drawing shown is of a shoebox with its lid. There are three views of the box that can be seen simultaneously. An isometric drawing does not show perspective. In

ISOSCELES TRIANGLE

267

other words, lines that are the same length do not appear shorter when they are further from your eye.

Reference: Elevation.

ISOMETRIC GRAPH PAPER

Reference: Isometric.

ISOMETRY

References: Congruent Figures, Transformation Geometry.

ISOSCELES TRAPEZIUM

This is a trapezium that has two sides equal in length and two pairs of congruent angles (see figure). It has one axis of symmetry and rotational symmetry of order one. An isosceles trapezium is what is left when an isosceles triangle is removed from a larger isosceles triangle, as shown in the figure. This removal concept is also true for equilateral triangles. In the figure the trapeziums are drawn in an upright position, but they may take up a variety of positions due to rotations and reflections.

References: Congruent Figures, Equilateral Triangle, Isosceles Triangle, Symmetry, Trapezium.

ISOSCELES TRIANGLE

This is a triangle with two equal sides and two equal angles. The two equal angles are often called the base angles of the isosceles triangle. A geometry theorem is stated as:

Theorem. The base angles of an isosceles triangle are equal.

All isosceles triangles have one axis of symmetry and rotational symmetry of order one. A variety of isosceles triangles are drawn in figure a.

268 ISOSCELES TRIANGLE

Acute angled isosceles triangle

Obtuse ang led Ri gh t an gl ed

isosceles triangle isosceles triangle

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