Functions

Representations of FunctionsThere 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

A rA r

2 rA r

A A r

B

C

D

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.

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.

xyxy

xyx

Classification of FunctionsWe 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

Function’s 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

Power Functions

Exponential Functions

Inverse Functions

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 don’t have inverse functions. The difficulty is overcome by restricting the domains of these functions so that hey become one-to-one.

Inverse Trigonometric Functions