Any questions on the Section 3.1 homework?PowerPoint PresentationSection 3.2 Introduction to FunctionsSlide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Reminder:Slide 31Any questions on the Section 3.1 homework?Turn in your worksheet for the 3.1 homework now.Now pleaseCLOSE YOUR LAPTOPSand turn off and put away your cell phones.Sample Problems Page Link(Dr. Bruce Johnston)Section 3.2Introduction 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.Find the domain and range of the relation {(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.ExampleFind the domain and range of the relation {(4,9), (-4,9), (2,3), (10,-5)}.Find the domain and range of the relation {(4,9), (-4,9), (2,3), (10,-5)}.Input (Animal)•Polar Bear•Cow•Chimpanzee•Giraffe•Gorilla•Kangaroo•Red FoxOutput (Life Span) 20 15 10 7ExampleFind the domain and range of the following relation.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.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 (y-values) , 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).ExampleIs 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?Example8642-2-4-6-10-5510fx  = x2-xIs 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?Example•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.ExampleUse the vertical line test to determine whether the graph to the right is the graph of a function.xySince no vertical line will intersect this graph more than once, it is the graph of a function.ExampleUse the vertical line test to determine whether the graph to the right is the graph of a function.xySince no vertical line will intersect this graph more than once, it is the graph of a function.ExampleUse the vertical line test to determine whether the graph to the right is the graph of a function.Since a vertical line can be drawn that intersects the graph at every point, it is NOT the graph of a function.xySince 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.ExampleUse the vertical line test to determine whether the graph to the right is the graph of a function.Since vertical lines can be drawn that intersect the graph in two points, it is NOT the graph of a function.xyFind the domain and range of the function graphed (in red) to the right. Use interval notation.xyDomain is [-3, 4]DomainRange is [-4, 2]RangeDetermining the domain and range from the graph of a relation:Example:ExampleFind the domain and range of the function graphed to the right. Use interval notation.xyDomain is (-, )DomainRange is [-2, )RangeExampleFind the domain and range of the function graphed to the right. Use interval notation.xyDomain: (-, )Range: (-, )ExampleFind the domain and range of the function graphed to the right. Use interval notation.xyDomain: (-, )Range: [-2.5](The range in this case consists of one single y-value.)ExampleFind 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]xyExampleFind 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: (-, )xyProblem 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).ExampleGiven that g(x) = x2 –