1. Functions

3. 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

4. B C

5. 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.

6. 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.

7. 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

8. 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

9. Symmetry

10. Transformations of Functions

11. Combinations of Functions

12. Composition of Functions

15. Power Functions

16. Exponential Functions

17. Inverse Functions

20. Logarithmic Functions

21. The logarithm with base is called the natural logarithm and has a special notation :

22. 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