2.3 Domain and Range of a Function


 Cornelius Adams
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1 Section Domain and Range o a Function Domain and Range o a Function Functions Recall the deinition o a unction. Deinition 1 A relation is a unction i and onl i each object in its domain is paired with one and onl one object in its range. This is not an eas deinition, so let s take our time and consider a ew eamples. Consider the relation R. R = { (0, 1), (0, 2), (3, 4) } The domain is {0, 3} and the range is {1, 2, 4}. Note that the number 0 in the domain o R is paired with two numbers rom the range, namel, 1 and 2. Thereore, R is not a unction. There is a construct, called a mapping diagram, which can be helpul in determining whether a relation is a unction. To crat a mapping diagram, irst list the domain on the let, then the range on the right, then use arrows to indicate the ordered pairs in our relation, as shown in Figure 1. 0 R Figure 1. A mapping diagram or R. It s clear rom the mapping diagram in Figure 1 that the number 0 in the domain is being paired (mapped) with two dierent range objects, namel, 1 and 2. Thus, R is not a unction. Let s look at another eample. Eample 1 Is the relation described in T a unction? First, the listing o the relation T. T = { (1, 2), (3, 2), (4, ) } Net, construct a mapping diagram or the relation T. List the domain on the let, the range on the right, then use arrows to indicate the pairings, as shown in Figure 2.
2 2 Chapter T 2 4 Figure 2. A mapping diagram or T. From the mapping diagram in Figure 2, we can see that each domain object on the let is paired (mapped) with eactl one range object on the right. Hence, the relation T is a unction. This is a good point to summarize what we ve reviewed about unctions thus ar. Summar 1 A unction consists o three parts: 1. a set o objects which mathematicians call the domain, 2. a second set o objects which mathematicians call the range, 3. and a rule that describes how to assign a unique range object to each object in the domain. The rule can take man orms. Function Notation Mathematicians are ond o the ollowing notation or unctions. () = 2 2. Now, let s see how this notations operates on the number. Thus, To ind where sends, we substitute or as ollows. () = 2 2. () = () 2 2(). Simpliing, () = 1. This result is read aloud as o equals 1, but we want to be thinking sends to 1. Let s look at eamples that use this notation. Eample 2 Given () = , determine ( 2).
3 Section Domain and Range o a Function 3 Simpl substitute 2 or. That is, ( 2) = ( 2) 3 + 3( 2) 2 = 8 + 3(4) = = 1. Thus, ( 2) = 1. Again, even though this is pronounced o 2 equals 1, we still should be thinking sends 2 to 1. Eample 3 Given determine (6). () = + 3 2, Simpl substitute 6 or. That is, (6) = (6) = 9 12 = 9 7. Thus, (6) = 9/7. Again, even though this is pronounced o 6 equals 9/7, we should still be thinking sends 6 to 9/7. Eample 4 Given () = 3, determine (a + 2). Note that this is again a simple substitution, where we replace each occurrence o in the ormula () = 3 with the epression a + 2. (a + 2) = (a + 2) 3. Finall, use the distributive propert to irst multipl b, then subtract 3. (a + 2) = a = a + 7. A good trick to keep in mind is that unction notation also gives the three parts o a unction. In () = notice that denotes the name o the rule, is the value rom the domain, and is the value mapped to in the range.
4 4 Chapter 2 Linear Functions Recall that the deinition o a linear unction is directl related to the slope intercept orm o a line. The SlopeIntercept Form o a Line. This orm o the equation o a line is called the slopeintercept orm. unction deined b the equation The () = m + b is called a linear unction. We start our review o how to determine the domain and range o a unction rom the graph with linear unctions. The Domain and Range o a Function Graphicall We can use the graph o a unction to determine its domain and range. For eample, consider the graph o the unction shown in Figure 3(a). P Q 3 4 (a) (b) (c) Figure 3. Determining the domain o a unction rom its graph. To determine the domain, we must collect the values (irst coordinates) o ever point on the graph o. In Figure 3(b), we ve selected a point P on the graph o, which we then project onto the ais. The image o this projection is the point Q, and the value o the point Q is an element in the domain o. Now, to ind the domain o the unction, we must project each point on the graph o onto the ais. Here s the question: i we project each point on the graph o onto the ais, what part o the ais will lie in shadow when the process is complete? The answer is shown in Figure 3(c). In Figure 3(c), note that the shadow created b projecting each point on the graph o onto the ais is shaded in red. This collection o values is the domain o the unction. There are three critical points that we need to make about the shadow on the ais in Figure 3(c).
5 Section Domain and Range o a Function 1. All points ling between = 3 and = 4 have been shaded on the ais in red. 2. The let endpoint o the graph o is an open circle. This indicates that there is no point plotted at this endpoint. Consequentl, there is no point to project onto the ais, and this eplains the open circle at the let end o our shadow on the ais. 3. On the other hand, the right endpoint o the graph o is a illed endpoint. This indicates that this is a plotted point and part o the graph o. Consequentl, when this point is projected onto the ais, a shadow alls at = 4. This eplains the illed endpoint at the right end o our shadow on the ais. We can describe the values o the shadow on the ais using setbuilder notation. Domain o = { : 3 < 4}. Note that we don t include 3 in this description because the let end o the shadow on the ais is an empt circle. Note that we do include 4 in this description because the right end o the shadow on the ais is a illed circle. We can also describe the values o the shadow on the ais using interval notation. Domain o = ( 3, 4] To ind the range o the unction, picture again the graph o shown in Figure 4(a). Proceed in a similar manner, onl this time project points on the graph o onto the ais, as shown in Figures 4(b) and (c). Q P 4 2 (a) (b) (c) Figure 4. Determining the range o a unction rom its graph. Note which part o the ais lies in shadow once we ve projected all points on the graph o onto the ais. 1. All points ling between = 2 and = 4 have been shaded on the ais in red (a thicker line stle i ou are viewing this in black and white).
6 6 Chapter 2 2. The let endpoint o the graph o is an empt circle, so there is no point to project onto the ais. Consequentl, there is no shadow at = 2 on the ais and the point is let unshaded (an empt circle). 3. The right endpoint o the graph o is a illed circle, so there is a shadow at = 4 on the ais and this point is shaded (a illed circle). We can now easil describe the range in both setbuilder and interval notation. Range o = ( 2, 4] = { : 2 < 4} Let s look at another eample. Eample Use setbuilder and interval notation to describe the domain and range o the unction represented b the graph in Figure (a). 4 (a) (b) Figure. Determining the domain rom the graph o. To determine the domain o, project each point on the graph o onto the ais. This projection is indicated b the shadow on the ais in Figure (b). Two important points need to be made about this shadow or projection. 1. The let endpoint o the graph o is empt (indicated b the open circle), so it has no projection onto the ais. This is indicated b an open circle at the let end (at = 4) o the shadow or projection on the ais. 2. The arrowhead on the right end o the graph o indicates that the graph o continues downward and to the right indeinitel. Consequentl, the projection onto the ais is a shadow that moves indeinitel to the right. This is indicated b an arrowhead at the right end o the shadow or projection on the ais. Consequentl, the domain o is the collection o values represented b the shadow or projection onto the ais. Note that all values to the right o = 4 are shaded on the ais. Consequentl,
7 Section Domain and Range o a Function 7 Domain o = ( 4, ) = { : > 4}. To ind the range, we must project each point on the graph o (redrawn in Figure 6(a)) onto the ais. The projection is indicated b a shadow or projection on the ais, as seen in Figure 6(b). Two important points need to be made about this shadow or projection. 3 (a) (b) Figure 6. Determining the range rom the graph o. 1. The let endpoint o the graph o is empt (indicated b an open circle), so it has no projection onto the ais. This is indicated b an open circle at the top end (at = 3) o the shadow on the ais. 2. The arrowhead on the right end o the graph o indicates that the graph o continues downward and to the right indeinitel. Consequentl, the projection o the graph o onto the ais is a shadow that moves indeinitel downward. In Figure 6(b), note how projections o points on the graph o not visible in the viewing window come in rom the lower right corner and cast shadows on the ais. Consequentl, the range o is the collection o values shaded on the ais o the coordinate sstem shown in Figure 6(b). Note that all values lower than = 3 are shaded on the ais. Thus, the range o is Range o = (, 3) = { : < 3}.
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