Chapter Two Functions and their Graphs yfx y

Chapter Two Functions and their Graphs • • y=f(x) y equals f of x f represents the function x called the independent variable y called the dependent variable → output value of f at x

Definition 2. 1. 1 A function f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B

Domain and Rage of function Input (domain) f Output (range) Function as a kind of machine

Remark 1. 2. 3. 4. Each element of A must be matched with an element of B Some elements of B may not be matched with any element of A Two or more element of A may be matched with the same element of B An element of A cannot be matched with two different element of B

Domain and Rage of function • Natural domain of function Df All real numbers for which the formula makes sense and function has a real value • Range of the function Rf The set of All values of f(x). if the range is real → the function is called real value

Ex. Find the domain of the function The domain is the set of all x-valued for which Then the domain of f(x) is Df=[-2 , 2]

Ex. Find the range of the function To find the range we have the domain of f(x) is Df=[-2 , 2]

Ex. Find the domain and range of the function

Remark • The range of the function is 0≤y<∞ • And the range of the function is a≤y<∞

Ex. 5 ( Convention on Domains) Find the domain of each of the following function

Graphs of function Graph of f={(x , y) : x element of D and y=f(x)} For sketching the graph of function f(x) where x element of Df. We plot enough points so that we can “see” what the graph is cellar

Piecewise function This function is defined by different formulas on different parts of their domain Ex. 12 Sketch the graph of X -3 -1 0 1 2 3 4 Y -5 -1 0 1 4 9 16 (1, 1) (2, 4) (x, y) (-3, -5) (-1, -1) (0, 0)

The graph of function

Test Functions Graph 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

Test Functions Graph

Summary of standard curves

Summary of standard curves

Summary of standard curves

Ex. Find the domain and range of function

Exercises 2. 1. Find the domain of function

Find the domain of function