9.3 Graphing Functions by Plotting Points, The Domain and Range of Functions

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1 9. Graphing Functions by Plotting Points, The Domain and Range o Functions Now that we have a basic idea o what unctions are and how to deal with them, we would like to start talking about the graph o a unction. First, the ollowing deinition, Deinition: The graph o a unction is the set o all points o the orm ( ( ) ) values are in the domain o.,, where the So, the most undamental way in which we can graph a unction is the same as the most undamental way we can graph anything, that is, by plotting points. Beore we get to that, recall the ollowing: Domain o a unction set o all irst coordinates, Range o a unction set o all second coordinates So that means the domain is the set o -values that can be graphed, and the range is the () or y-values that get graphed. Using these deinitions and concepts we can begin to graph unctions and determine their domains and ranges. Eample 1: Graph ( ) + 1. Determine the domain and range. First we need to construct a table o values. We can choose any values we would like that will return a real number. Since our unction has a radical we will have to be very careul how we choose our values. It should be clear that i we take any value smaller than 1 we will end up with a negative under the radical. Which is not a real number. So lets start at 1 and take larger values. X () ( 1 ) 1+ ( 1) 0 0 ( ) 1+ ( ) 1. 7 ( 0 ) 1+ ( 0) 1 1 ( ) 1+ ( ) 4 () 1 1+ () We can now plot these points and graph the curve. I this is not enough points to tell what the graph looks like, we can always plot more.

2 Now, using the graph we can determine the domain and range o the unction. From above, we know that the domain is the -values. So by the graph, we can see that the only -values that have a y-value associated with them are rom 1 on. So as an interval we write that as [ 1, ). Likewise or the range we look at the y-values that get used. We can see that all the y-values 0,. rom 0 upwards would get used. We write this in interval notation as [ ) Eample : Graph ( ) 1. Determine the domain and range. Again we will plot several points to get an idea what the graph o the unction is doing. I its not enough we can always plot more. We usually like to start with easy values. X () ( ) ( ) ( 1 ) ( 1) ( 1 ) ( 1) 1 ( ) ( ) 1 ( 0 ) ( 0) I we plot these points we ind that we cannot tell what happens on the graph between 0 and 1. So lets plot another point say 1. X () 1 0 ( 1 1 ) ( ) So plotting these points we get the ollowing In general, graphs o unctions that contain an absolute value will have at least one sharp corner like the one seen above. Again the domain o the unction is the -values, we can clearly see all -values will get used. As an interval we write (, ). For the range, we look at the used y-values and we can see that 0, ) we have rom 0 up. That is, [. Eample : Graph ( ) 1. Determine the domain and range.

3 Again we will plot several points. X () ( 1) ( 1) ( 1) ( 1) ( 0) ( 0) ( 0) ( 0) Plotting these points we get. ( ) ( ) ( ) ( ) 11 () () () ( ) ( ) ( ) ( ) ( ) Finally, we can see that the all - and y-values will be used in this unction. Thus, the domain and ranges are both. (, ) As it turns out not every graph represents a unction. Since, in order to have a unction, one value o into the unction must give us only one () value out o the unction we get the ollowing test or unctions. The Vertical Line Test A graph deines a unction i any vertical line intersects the graph at no more than one point. What that means is in order to see i a graph represents a unction, we simply need to see i any vertical lines hit the graph in more than one point. I it does, the graph is not a unction. Eample 4: Determine i the graph represents a unction. I it is a unction, determine the domain and range. a. b. c.

4 Parts a and b both pass the vertical line test since every possible vertical line will hit at most one point. Thus a and b are unctions. However, take almost any vertical line through part c. You will get the line hitting two points. Thus, part c ails the vertical line test and is thereore not a unction. Lastly, we want to be able to determine the domain o a unction without looking at its graph. This is a relatively simple task. Since the domain is the values that we can put into a unction to get a real number out, all we have to do is eliminate all the possible values that would give us a non-real number. That is, get rid o any numbers which give us a negative under the radical (this gives us comple numbers) and which give us a zero in the denominator (this is undeined). Taking care o these two cases will be suicient or now. However, in the uture, some unctions we will look at in this course will also have to have a restricted domain. We will see these unctions in Chapter 11. Finding the Domain o a Function To ind the domain o a unction, just take care o the ollowing two cases: 1. We cannot have a negative under the radical.. We cannot have any zeros in the denominator. The domain is all real numbers ecept where either o the preceding occur. Eample 5: Find the domain o the ollowing. Put your answer in interval notation. 6 a. ( ) b. g ( ) 5 a. We are only worried about zeros in the denominator or negatives under the radical. In this case, there is no radical, so we only need to know when the denominator is zero. ( ) is undeined eactly when 6 0. So we solve ( ) ( + ) 0 (By actoring) 0 or + 0 So ( ) is undeined when and. 6 Thus our domain is all real numbers ecept,., U, U,. In interval notation: ( ) ( ) ( ) b. Again we want to eliminate zeros rom the denominator and negative under radicals. But in this case, we have no denominator to worry about. So we want no negatives under the radical. However, that is the same as saying:

5 g is when 5 0. I the portion under the radical is greater or equal to zero, it is certainly not negative. The domain o ( ) 5 So we solve the inequality as beore Thus, the domain o g ( ) 5 is [, ) Eercises Graph the ollowing unctions. Determine the domain and range o each. 1. ( ) 1. g( ) h ( ) ( ) 4 5. ( ) 6. g( ) 1 7. h ( ) F ( ) + 9. ( ) 10. h ( ) ( ) 1. g ( ) ( ) ( ) + 1 g 15. ( ) g ( ) h 17. ( ) 18. p ( ) q ( ) ( ) ( ). h ( ) ( ) 1 g 4. ( ) 5. r( ) 6. S ( ) ( ) 1 g 1 4) 8. ( ) ( 9. h ( ) + 0. g ( ) ( 1)( + 1)( ) 1. Graph the ollowing unctions. a. ( ) b. ( ) ( ) c. ( ) d. ( ) e. ( ). ( ) ( ) What is the relationship between parts a. and b.? a. and c.? a. and d.? a. and e.? a. and.?. Graph the ollowing unctions. a. g ( ) b. g ( ) + c. g ( ) + d. g( ) e. g ( ) +. g ( ) + + What is the relationship between parts a. and b.? a. and c.? a. and d.? a. and e.? a. and.? Determine i the ollowing graph represents a unction.. 4.

6 Find the domain in interval notation o the ollowing unctions. 41. h ( ) 1 4. ( ) 4 4. ( ) g ( ) ( ) h ( ) 1 t g() t t + t 48. ( ) h ( ) ( ) g( ) 0 t 5. () t t t 5 5. ( ) 54. g ( ) ( ) g( ) 1 l 57. h () l l ( ) 59. g( ) g ( ) 61. h ( ) 6. ( ) 6. ( )