Transcript Chapter 1
Functions 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 A of a circle depends on the radius r of the circle. The rule that connects A and r is given by the equation: A r 2 With each positive number r there is associated one value of A, and we say that A is a function of r 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 xy-plane. But the question arises: Which curves in the xy -plane are graphs of functions? This is answered by the following test. The Vertical Line Test A curve in the xy-plane is the graph of a function of x 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 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: Inverse Trigonometric Functions 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.