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Functions

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Representations of Functions

- There are four possible ways to represent a

function - verbally (by a description in words)
- numerically (by a table of values)
- visually (by a graph)
- algebraically (by an explicit formula)
- A The area of a circle depends on the

radius of the circle. The rule that connects

and is given by the equation - With each positive number there is

associated one value of , and we say that

is a function of

B

C

The rule that the U. S. Postal Service used as of

2001 is as follows The cost is 34 cents for up

to one ounce, plus 22 cents for each successive

ounce up to 11 ounces.

D

Graphs of Functions

- The graph of a function is a curve in the

-plane. But the question arises Which curves in

the -plane are graphs of functions? This is

answered by the following test. - The Vertical Line Test A curve in the

-plane is the graph of a function of if and

only if no vertical line intersects the curve

more than once.

Classification of Functions

- We may classify functions by their formula as

follows - Polynomials
- Linear Functions, Quadratic Functions.

Cubic Functions. - Piecewise Defined Functions
- Absolute Value Functions, Step Functions
- Rational Functions
- Algebraic Functions
- Trigonometric and Inverse trigonometric

Functions - Exponential Functions
- Logarithmic Functions

Functions Properties

- We may classify functions by some of their

properties as follows - Injective (One to One) Functions
- Surjective (Onto) Functions
- Odd or Even Functions
- Periodic Functions
- Increasing and Decreasing Functions
- Continuous Functions
- Differentiable Functions

Symmetry

Transformations of Functions

Combinations of Functions

Composition of Functions

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Power Functions

Exponential Functions

Inverse Functions

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Logarithmic Functions

The logarithm with base is called the natural

logarithm and has a special notation

When we try to find the inverse trigonometric

functions, we have a slight difficulty. Because

the trigonometric functions are not one-to-one,

they dont have inverse functions. The difficulty

is overcome by restricting the domains of these

functions so that hey become one-to-one.

Inverse Trigonometric Functions

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