Turn in your worksheet for the 3 1 homework now Any questions on the Section 3 1 homework Now please CLOSE YOUR LAPTOPS and turn off and put away your cell phones Sample Problems Page Link Dr Bruce Johnston Section 3 2 Introduction to Functions Equations in two variables define relations between the two variables There are other ways to describe relations between variables Set to set Ordered pairs A set of ordered pairs x y is also called a relation The domain is the set of x coordinates of the ordered pairs The range is the set of y coordinates of the ordered pairs Example Find the of of thethe relation Find thedomain domainand andrange range relation 4 9 2 3 10 5 4 9 4 9 4 9 2 3 10 5 Domain is the set of all x values 4 4 2 10 Range is the set of all y values 9 3 5 Note if an element number is repeated it only appears in the list one time Example Find the domain and range of the following relation Input Animal Polar Bear Cow Chimpanzee Giraffe Gorilla Kangaroo Red Fox Output Life Span 20 15 10 7 Example cont Domain Polar Bear Cow Chimpanzee Giraffe Gorilla Kangaroo Red Fox Range 20 15 10 7 Some relations are also functions A function is a set of order pairs in which each first component in the ordered pairs corresponds to exactly one second component Example Given the relation 4 9 4 9 2 3 10 5 is it a function Since each element of the domain x values is paired with only one element of the range yvalues it is a function Note It s okay for a y value to be assigned to more than one x value but an x value cannot be assigned to more than one y value if the relation is a function Each x value has to be assigned to ONLY one y value Example Is the relation y x2 2x a function Since each element of the domain the x values would produce only one element of the range the y values it is a function Question What does the graph of this function look like 8 6 4 2 f x x2 x 10 5 5 2 4 6 10 Example Is the relation x2 y2 9 a function Since each element of the domain the x values would correspond with 2 different values of the range both a positive and negative y value the relation is NOT a function Check the ordered pairs 0 3 0 3 The x value 0 corresponds to two different y values so the relation is NOT a function Question What does the graph of this relation look like Relations and functions can also be described by graphing their ordered pairs Graphs can be used to determine if a relation is a function If an x coordinate is paired with more than one y coordinate a vertical line can be drawn that will intersect the graph at more than one point If no vertical line can be drawn so that it intersects a graph more than once the graph is the graph of a function This is the vertical line test Example Use the vertical line test to determine whether the graph to the right is the graph of a function y x Since no vertical line will intersect this graph more than once it is the graph of a function Example Use the vertical line test to determine whether the graph to the right is the graph of a function y x Since no vertical line will intersect this graph more than once it is the graph of a function Example Use the vertical line test to determine whether the graph to the right is the graph of a function y x Since a vertical line can be drawn that intersects the graph at every point it is NOT the graph of a function Since the graph of a linear equation is a line all linear equations are functions except those whose graph is a vertical line An equation of the form x c is a vertical line and IS NOT a function Example Use the vertical line test to determine whether the graph to the right is the graph of a function y x Since vertical lines can be drawn that intersect the graph in two points it is NOT the graph of a function Determining the domain and range from the graph of a relation Example y Find the domain and range of the function graphed in red to the right Use interval notation Domain x Domain is 3 4 Range is 4 2 Range Example Find the domain and range of the function graphed to the right Use interval notation Domain is Range is 2 y Range x Domain Example Find the domain and range of the function graphed to the right Use interval notation y x Domain Range Example Find the domain and range of the function graphed to the right Use interval notation y x Domain Range 2 5 The range in this case consists of one single y value Example Find the domain and range of the relation graphed to the right Use interval notation Note this relation is NOT a function but it still has a domain and range Domain 4 4 Range 4 3 0 y x Example Find the domain and range of the relation graphed to the right Use interval notation Note this relation is NOT a function but it still has a domain and range Domain 2 Range y x Problem from today s homework Answer Domain is 3 1 0 2 3 Range is 3 2 This relation IS a function Specialized notation is often used when we know a relation is a function and it has been solved for y For example the graph of the linear equation y 3x 2 passes the vertical line test so it represents a function We often use letters such as f g and h to name functions We can use the function notation f x and write the equation as f x 3x 2 Note The symbol f x read f of x is a specialized notation that does NOT mean f x f times x When we want to evaluate a function at a particular value of x we substitute the x value into the notation For example f 2 means to evaluate the function f when x 2 So we replace x with 2 in the equation For our previous example when f x 3x 2 f 2 3 2 2 6 2 4 When x 2 then f x 4 giving us the ordered pair 2 4 Example Given that g x x2 2x find g 3 Then write down the corresponding ordered pair g 3 3 2 2 3 9 6 15 The ordered pair is 3 15 Example Given the graph of the following function find each function value by inspecting the graph f 0 2 f 4 3 f 5 1 f 6 7 y f x x …