1. 1.Definition of a function 2.Finding function values 3.Using the vertical line test

2. Function: A function f is a correspondence from a set D to a set E that assigns to each element x of D exactly one value ( element ) y of E Graphical Illustration E x * z * w * 5 * * f(w) * f(x) * f(z) * f(5) * 3 * 4 * - 9 D f f is a function

3. More illustrations…. x * z * w * 5 * * f(w) * f(x) * f(z) * f(5) * 3 * 4 * - 9 D E f is not a function Why? x in D has two values x * z * w * 5 * * f(w) * f(x) * f(z) * f(5) * 3 * 4 * - 9 D E f is not a function Why? x in D has no values

4. How to find function values? Example 1: Let f be the function with domain R such that f( x) = x 2 for every x in R. ( i ) Find f ( -6 ), f ( ), f( a + b ), and f(a) + f(b) where a and b are real numbers. Solution: Note: f ( a + b ) f( a ) + f ( b )

5. Vertical Line Test of functions Vertical Line test: The graph of a set of points in a coordinate plane is the graph of a function if every vertical line intersects the graph in at most one point Example: check if the following graphs represent a function or not Function Not Function

6. Increasing, Decreasing and Constant Functions over an interval I f(x 1 ) = f(x 2 ) whenever x 1 x 2 f is constant over interval I f(x 1 ) > f(x 2 ) whenever x 1 < x 2 f is decreasing over interval I f(x 1 ) < f(x 2 ) whenever x 1 < x 2 f is increasing over interval I Graphical InterpretationDefinitionTerminology x1x1 x2x2 f(x 1 ) f(x 2 ) x y x1x1 x2x2 f(x 1 ) f(x 2 ) x y x1x1 x2x2 f(x 1 )f(x 2 ) x y Interval I