Chapter 1:

Functions

Chapter 2:

Limits

Chapter 3:

Continuity

Objectives:

Work with functional notations and use them to

express the relation between variables.

Define the domain and determine the range of a

function.

Draw the graph of function.

Perform operations on functions

Function -- is a rule that assigns to each element X in set A

exactly one element called f(x) in set B.

Four ways to represent functions

1. Verbally: The Circumference of a circle increases with its radius.

2. Numerically: Table Values

Quiz Score

1 95

2 98

3. Graphically

4. Algebraically (Use of explicit formula)

HOW FUNCTIONS ARE DEFINED:

a. Explicitly: y = f (x)

y = x2 –x +2

b. Implicitly: f (x, y) = 0

x2 + xy2 = 9

c. Parametric form: x = f(t), y = g(t)

* t = parameter

x = 2t – 1 y = t2

SYMMETRY:

The function is an EVEN function if f (x) = f (-x).

The graph of the function is symmetric with

respect to the vertical axis. (E.g. y = x2 )

The function is an ODD function if f (- x) = - f (x).

The graph of the function is symmetric with

respect to the origin. (E.g. y = x3 )

CLASSIFICATION OF FUNCTIONS:

1. Algebraic functions

2. Transcendental functions

a. trigonometric

b. inverse trigonometric

c. exponential

d. logarithmic

e. hyperbolic

“Domain and Range of a function”

Set A is the domain of the function, while x is the independent

variable.

Set B is the range of the function, while f (x) is the dependent

variable.

Domain: { x | -2 x 2 }

Range: { f (x) | 0 f (x) 2 }

*Restriction is necessary for f (x) to be real.

Example 1: Find the natural domain and range of :

Example 2: Find the natural domain and range of :

Example 3: Find the natural domain and range of :

Example 4: Find the natural domain of:

Example 5: Find the natural domain and range of:

Example 6: Find the natural domain of:

“Piecewise Defined Function”

The function has different explicit formulas in different

intervals of its domain.

Example 1: Draw the graph of:

Example 2: Draw the graph of:

Example 3: Draw the graph of:

Example 4: Draw the graph of:

Example 5: Draw the graph of:

“Functions as Mathematical Models of Reality”

Mathematics can be used as a basis for decision

making. Many situations in real life can be

represented with a mathematical model, usually

as functions.

Example 1: Let P be the perimeter of an equilateral

triangle. Write a formula A(P), the area of thetriangle as a function of the perimeter.

Example 2: Express the area of a circle as a function

of its circumference.

Example 3: Express the volume of a sphere as a

function of its (a) diameter (b) surface area

Example 4: Express the surface area of a cube as a

function of its volume.

Example 5: A rectangle has a perimeter of 20 inches.

Express the area as a function of one of itssides.

Example 6: Write a formula describing the distance

of a point on the parabola x = 2y2 to (10,0) as afunction of x.

Example 7: One of the legs of a right triangle has a

length of 4 cm. Express the length of thealtitude perpendicular to the hypotenuse as afunction of the length of the hypotenuse.

Example 8: Boxes are to be made from rectangular

cardboards, 8 inches by 15 inches. Equalsquares are to be cut from the four corners,then the flaps are folded upward. Express thevolume of the box as a function of a side ofthe squares from the corners.