Real Numbers

Functions and Inverse Functions

Composite Functions

Inverse Composite Functions

Some Worked Problems on Indices

Formulae for regular polygons

Some similar solids

Natural numbers, denoted by N, are all the positive numbers from 1 onward. 1 is the smallest natural number. Natural numbers extend to + infinity. Thus the set N = {1, 2, 3, ...}

Whole numbers include 0 along with the set N. 0 is the smallest whole number. The set W = {0, 1, 2, 3, ...} to + infinity.

The set of integers includes 0, and the negative numbers. The set Z of integers extends to ± infinity: Z = {... -3, -2, -1, 0, 1, 2, 3, ...}. Z contains 3 disjoint sets: {Positive integers}, {Negative integers} and {0}.

Rational numbers are numbers that can be expressed as the quotient of 2 integers a and b provided b doesn't = 0. All integers are rational as all are divisible by 1. Repeating and terminating decimals are rational, e.g. 1.3333... and 0.123. The set of rational numbers is denoted Q.

Surds, or irrational numbers, are non-repeating infinite numbers such as pi. pi is the ratio of the circumference of a circle to its diameter in radians. 3.1416 is a rational approximation to 4 decimal places. Other examples of surds include root3, pi/4, 4 - 2 root11. The set of surds is denoted by J.

Real numbers, R, are all rational or irrational numbers. (They exclude complex numbers such as root-1, denoted by i).

Complex numbers, denoted by z, are numbers in the form a + ib, where i = root -1 and a and b are real numbers. a is the real part, Rz, and b, the imaginary part Iz. e.g. z = 3 + 4i, where 3 = Rz, 4 = Iz.

If a given value of x has only 1 corresponding y value, y is a function of x, e.g. y = 2x + 7, and to denote the function of x, f(x) = 2x + 7. x values are referred to as the domain and y values are referred to as the range. Each pair of corresponding values for x and y constitute an ordered pair. An equation such as y = x

Ways to represent functions:

- Mapping diagrams

- Algebraic equations

- Graphs

- Sets of ordered pairs of corresponding x & y values

Flow charts may also be used to show a relation between x and y, e.g. f(x) = 3(2x + 5), x = 7:

x. Enter x value (7)

2x. Multiply by 2 (14)

2x + 5. Add 5 (19)

3(2x + 5). Multiply by 3 (57)

f(x) = 3(2x + 5). Read f(x) (y = 57)

To find the inverse of a function, simply interchange the range values with the domain values for x and y. The inverse of f(x) is denoted by f^{ -1}(x).

e.g. f(x) = x + 4/ 5, the inverse function is found by reversing the order of operations:

f^{ -1}(x) = 5x - 4. The inverse of a function can help in the solution of equations.

e.g. f(x) = (x - 4) / (3 - x), then y = (x - 4) / (3 - x). And for f^{ -1}(x): x = (y - 4) / (3 - y), we must rearrange making y the subject:

x (3 - y) = y - 4

3x - xy = y - 4 *Multiplying out ()'s*

-y - xy = -3x - 4 *Swap sides gives*

3x + 4 = y +xy *and isolating y*

3x + 4 = y(1 + x) *obtaining y by swapping sides*

(3x + 4) / (1 + x) = y

We can now find specific values for y, e.g. f^{ -1}(2):

(3 x 2 + 4) / (1 + 2) = 10/3, and y = 10/3.

e.g. f^{ -1}(-2):

(3 x (-2) + 4) / (1 + (-2)) = (-6 + 4) / -1 = -2 / -1 = 2, and y = 2.

Composite functions are a combination of different functions in an equation. e.g. f(x) = 2x - 1, g(x) = 3x + 2. Composite functions are seldom commutative, i.e. fg doesn't = gf:

fg (x) = 2 (3x + 2) - 1 = 6x + 3, solving g first.

gf (x) = 3(2x - 1) + 2 = 6x - 1, solve f first.

For f(x) = 2x + 5, g(x) = 0.5x, h(x) = 3x - 1, finding composite function fgh(x):

y = 2[0.5 {3x - 1}] + 5 = 3x + 4

If fgh(3) = 3 x 3 + 4 = 13, y = 13.

Find ghf(x) = 0.5 [3 (2x +5) - 1] = 3x + 7

e.g. ghf(-2) = 3 x (-2) + 7 = 1, y = 1.

To find the inverse composite function first obtain any unknown composite functions. Inverse composite functions are denoted by e.g. (fg)

e.g. Finding a composite function for gf if f(x) = 3x - 5 and g(x) = 2x + 1:

gf(x) = 2(3x - 5) + 1 = 6x - 9

And the inverse, (gf)

f

f

= 1/6 (x - 1) + 5/3

= 1/6x - 1/6 + 5/3

= 1/6x + 9/6

= 1/6 (x + 9)

(gf)

Thus, (gf)

f(x) = 7x + 2 and g(x) = 2x - 1, find (gf)^{ -1}

f^{ -1}(x) = 1/7 (x - 2) and g^{ -1}(X) = 1/2 (x + 1)

and (gf)^{ -1} = f^{ -1}g^{ -1}(X) = 1/7 [1/2 (x + 1)] - 2/7

= 1/14 (x + 1) - 2/7

= 1/14x + 1/14 - 2/7

= 1/14 (x - 3)

Clearing the 3 equations to single fractions.

a) Simplify with positive indices:

(2^{4} x 4^{2} x 3^{ -3} + 2^{ -2} x 3^{2} x 4^{-3}) / 9^{2} x 8^{3}.

Rewriting with positive indices and simplify to smallest possible bases:

(2^{4} x 2^{4} x 1/3^{3} + 1/2^{2} x 3^{2} x 1/2^{6}) / 3^{4} x 2^{9}.

Next, clear fractions in numerator:

3^{3} x [(2^{8} x 1/3^{3} + 3^{2} x 1/2^{8}) / 3^{4} x 2^{9}].

Leaves:

2^{8} x [(2^{8} + 3^{5} x 1/2^{8)} / 3^{7} x 2^{9}].

Giving the solution:

(2^{16} + 3^{5}) / (3^{7} x 2^{17}).

b) (3^{ -2}) x 5^{4} x 6^{ -7}) / (2/5)^{3} x (5/3)^{ -2}:

Clearing (3/5)^{ -2} by inverting (becomes (3/5)^{2} as positive indice), and multiply into equation instead of dividing:

5^{4} x 5^{3} x 5^{2} / 3^{2} x 2^{7} x 3^{7} x 2^{3} x 3^{2}

Simplifying to a solution with smallest possible bases:

= 5^{9} / 3^{11} x 2^{10}

c) {(7/8)^{ -2} - (1/4)^{4}} / {(1/16)^{2} x (1/32)^{ -3}}

Inversion to obtain positive indices gives:

{(8/7)^{2} - (1/4)^{4}} / (1/16)^{2} x 32^{3}

Simplifying bases and raising powers accordingly:

(2^{6}/7^{2}) - (1/2^{8}) / (1/2^{8}) x 2^{15}

Clear excess fractions top & bottom by multiplying by 2^{8} gives:

2^{8} x [(2^{6}/7^{2}) - (1/2^{8}) / (1/2^{8}) x 2^{15}]

Simplifying:

(2^{14}/7^{2} - 1) / 2^{15}

Clearing the fraction in the numerator by multiplying by 7^{2}:

7^{2} x [(2^{14}/7^{2} - 1) / 2^{15}]

Obtaining the final equation containing only 1 fraction:

(2^{14} - 7^{2}) / (2^{15} x 7^{2})

**Volume and area - similar solids:**

The surface area of similar solids are proportional to the squares of their linear dimensions. The volumes of similar solids are proportional to the cubes of their linear dimensions.

2 cylinders are similar:

Height of larger / height of smaller = Diameter of larger / diameter of smaller

Surface area of larger / surface area of smaller = (height of larger)^{2} / (height of smaller)^{2} = (Diameter of larger)^{2} / (diameter of smaller)^{2}

Volume of larger / volume of smaller = (Height of larger)^{3} / (height of smaller)^{3} = (Diameter of larger)^{3} / (diameter of smaller)^{3}

2 spheres are similar:

Surface area of smaller / surface area of larger = (Diameter of smaller)^{2} / (diameter of larger)^{2}

Surface area of the smaller sphere = area of larger sphere / denominator.

Volume of smaller / volume of larger = (Diameter of smaller)^{3} / (diameter of larger)^{3}

Volume of smaller sphere = Volume of larger / denominator.

**Formula for Regular Polygons:**

To find an unknown number of sides, n, from a given interior angle:

n = 360/(180-interior angle)

To find the unknown sum of interior angles, T from a given side:

T = (n-2) x pi radians = (n-2) x 180 ^{o}

To find an interior angle, t, from a given number of sides, n:

t = (180n - 360)/n = 180 - 360/n